diff --git a/dumux/discretization/staggered/freeflow/staggeredgeometryhelper.hh b/dumux/discretization/staggered/freeflow/staggeredgeometryhelper.hh
index 0a731d966f584b277498d6b1b7d9a6bcdde464b9..61948a2ac8b2fd3c558110734d18e7825906f856 100644
--- a/dumux/discretization/staggered/freeflow/staggeredgeometryhelper.hh
+++ b/dumux/discretization/staggered/freeflow/staggeredgeometryhelper.hh
@@ -39,6 +39,33 @@ namespace Detail {
 /*!
  * \ingroup StaggeredDiscretization
  * \brief Parallel Data stored per sub face
+ *
+ * ------------
+ * |          |
+ * |          |
+ * |          |
+ * -----------------------
+ * | yyyyyyyy s          |
+ * | yyyyyyyy s          |
+ * | yyyyyyyy s          |
+ * -----------------------
+ * In this corner geometry, hasParallelNeighbor will return true for subcontrolvolumeface s belonging to the
+ * element filled by 'y's, but hasParallelNeighbor will return false for the subcontrolvolumeface that has the
+ * same dofIndex. We name this situation hasHalfParallelNeighbor.
+ *
+ * ------------
+ * | yyyyyyyy s
+ * | yyyyyyyy s
+ * | yyyyyyyy s
+ * -----------------------
+ * |          |          |
+ * |          |          |
+ * |          |          |
+ * -----------------------
+ * In this corner geometry, hasParallelNeighbor will return true for subcontrolvolumeface s belonging to the
+ * element filled by 'y's. However, as there also might be a boundary velocity value known at the corner, which
+ * can be used instead of the standard parallel velocity in some cases, we want to identify this situation. We
+ * name it cornerParallelNeighbor.
  */
 template<class GridView, int upwindSchemeOrder>
 struct PairData
@@ -49,6 +76,8 @@ struct PairData
     using SmallLocalIndexType = typename IndexTraits<GridView>::SmallLocalIndex;
 
     std::bitset<upwindSchemeOrder> hasParallelNeighbor;
+    bool hasHalfParallelNeighbor = false;
+    bool hasCornerParallelNeighbor = false;
     std::array<GridIndexType, upwindSchemeOrder> parallelDofs;
     std::array<Scalar, upwindSchemeOrder> parallelCellWidths;
     bool hasOuterLateral = false;
@@ -378,6 +407,45 @@ private:
                     // recursively insert parallel neighbor faces into pair data
                     const auto parallelAxisIdx = directionIndex(intersection);
                     const auto localLateralIntersectionIndex = intersection.indexInInside();
+
+                   /*
+                    *       ------------
+                    *       |          |
+                    *       |          |
+                    *       |          |
+                    *       iiiiiiiiiii*bbbbbbbbbbb
+                    *       |          o zzzzzzzz |
+                    *       |          o zzzzzzzz |
+                    *       |          o zzzzzzzz |
+                    *       -----------------------
+                    *
+                    *       i:intersection,o:intersection_, b: outerIntersection, z: intersection_.outside()
+                    */
+                   if (intersection_.neighbor())
+                        for (const auto& outerIntersection : intersections(gridView_, intersection_.outside()))
+                            if (intersection.indexInInside() == outerIntersection.indexInInside())
+                                if (!outerIntersection.neighbor())
+                                    pairData_[numPairParallelIdx].hasHalfParallelNeighbor = true;
+
+                   /*       ------------
+                    *       |          o
+                    *       |          o
+                    *       |          o
+                    *       iiiiiiiiiii------------
+                    *       | zzzzzzzz b          |
+                    *       | zzzzzzzz b          |
+                    *       | zzzzzzzz b          |
+                    *       -----------------------
+                    *
+                    *       i:intersection,o:intersection_, b: outerIntersection, z: intersection.outside()
+                    */
+                   if (!intersection_.neighbor())
+                        for (const auto& outerIntersection : intersections(gridView_, intersection.outside()))
+                            if (intersection_.indexInInside() == outerIntersection.indexInInside())
+                                if (outerIntersection.neighbor())
+                                    pairData_[numPairParallelIdx].hasCornerParallelNeighbor = true;
+
+
                     addParallelNeighborPairData_(intersection.outside(), 0, localLateralIntersectionIndex, parallelLocalIdx, parallelAxisIdx, numPairParallelIdx);
                 }
 
diff --git a/dumux/discretization/staggered/freeflow/subcontrolvolumeface.hh b/dumux/discretization/staggered/freeflow/subcontrolvolumeface.hh
index 29de016a9ee8e9367d4717c4934c3b3e87186179..9805aa626ab4243d3dbc278c57dbaceee554deee 100644
--- a/dumux/discretization/staggered/freeflow/subcontrolvolumeface.hh
+++ b/dumux/discretization/staggered/freeflow/subcontrolvolumeface.hh
@@ -318,6 +318,53 @@ public:
         return pairData(localSubFaceIdx).hasParallelNeighbor[parallelDegreeIdx];
     }
 
+    /*!
+    * \brief Check if the face has a half parallel neighbor
+    *
+    * \param localSubFaceIdx The local index of the subface
+    *
+    * ------------
+    * |          |
+    * |          |
+    * |          |
+    * -----------------------
+    * | yyyyyyyy s          |
+    * | yyyyyyyy s          |
+    * | yyyyyyyy s          |
+    * -----------------------
+    * In this corner geometry, hasParallelNeighbor will return true for subcontrolvolumeface s belonging to the
+    * element filled by 'y's, but hasParallelNeighbor will return false for the subcontrolvolumeface that has the
+    * same dofIndex. We name this situation hasHalfParallelNeighbor.
+    */
+    bool hasHalfParallelNeighbor(const int localSubFaceIdx) const
+    {
+        return pairData(localSubFaceIdx).hasHalfParallelNeighbor;
+    }
+
+    /*!
+    * \brief Check if the face has a corner parallel neighbor
+    *
+    * \param localSubFaceIdx The local index of the subface
+    *
+    * ------------
+    * | yyyyyyyy s
+    * | yyyyyyyy s
+    * | yyyyyyyy s
+    * -----------------------
+    * |          |          |
+    * |          |          |
+    * |          |          |
+    * -----------------------
+    * In this corner geometry, hasParallelNeighbor will return true for subcontrolvolumeface s belonging to the
+    * element filled by 'y's. However, as there also might be a boundary velocity value known at the corner, which
+    * can be used instead of the standard parallel velocity in some cases, we want to identify this situation. We
+    * name it cornerParallelNeighbor.
+    */
+    bool hasCornerParallelNeighbor(const int localSubFaceIdx) const
+    {
+        return pairData(localSubFaceIdx).hasCornerParallelNeighbor;
+    }
+
    /*!
     * \brief Check if the face has an outer normal neighbor
     *
@@ -402,7 +449,7 @@ public:
     void setCenter(const GlobalPosition& center)
     { center_ = center; }
 
-    //! set the boudnary flag
+    //! set the boundary flag
     void setBoundary(bool boundaryFlag)
     { boundary_ = boundaryFlag; }
 
diff --git a/dumux/freeflow/navierstokes/staggered/staggeredupwindhelper.hh b/dumux/freeflow/navierstokes/staggered/staggeredupwindhelper.hh
index 27f69614645ed47033cfbeab6bbae5bb8bacb713..9d1a9996bd94ec7e3b954d792641e246ab173556 100644
--- a/dumux/freeflow/navierstokes/staggered/staggeredupwindhelper.hh
+++ b/dumux/freeflow/navierstokes/staggered/staggeredupwindhelper.hh
@@ -258,6 +258,40 @@ private:
      * TODO: In order to get a second order momentum upwind scheme for compressible flow the densities have to be evaluated
      * at the same integration points / positions as the velocities. The currently implementation just takes the closest upwind density
      * to compute the momentum as a crude approximation.
+     *
+     * ------------
+     * |     xxxx o
+     * |     xxxx o
+     * |     xxxx o
+     * -----------*bbbbbbbbbbb
+     * |     yyyy o zzzz     |
+     * |     yyyy o zzzz     |
+     * |     yyyy o zzzz     |
+     * -----------------------
+     * If scvf_ is touching a corner, at which there is a Dirichlet condition for face b (half-control
+     * volumes x, y and z), the transported velocity at * is given. No upwinding or
+     * higher-order approximation for the velocity at * are required. This also means the transported
+     * velocity at * is the same for the half-control volumes y and z.
+     *
+     * ------------
+     * |          |
+     * |          |
+     * |          |
+     * -----------------------
+     * |     xxxx o wwww     |
+     * |     xxxx o wwww     |
+     * |     xxxx o wwww     |
+     * ------+++++*~~~~~------
+     * |     yyyy o zzzz     |
+     * |     yyyy o zzzz     |
+     * |     yyyy o zzzz     |
+     * -----------------------
+     * If the flux over face + is calculated and a corner occurs a bit further away (here upper right),
+     * no special treatment of the corner geometry is provided. This means, the transported velocity at
+     * star (*) is obtained with an upwind scheme. In particularly, this also means
+     * that while x and y see the same velocity * for the flux over face x (continuity OK), z sees
+     * another velocity * for the flux over face ~ (continuity still OK, as only z and w have to use the
+     * same velocity at *).
      */
     std::array<Scalar, 3> getLateralSecondOrderUpwindMomenta_(const Scalar insideDensity,
                                                               const Scalar outsideDensity,
@@ -270,11 +304,22 @@ private:
 
         // If the lateral face lies on a boundary, we assume that the parallel velocity on the boundary is actually known,
         // thus we always use this value for the computation of the transported momentum.
-        if (!scvf_.hasParallelNeighbor(localSubFaceIdx, 0))
+        if (!scvf_.hasParallelNeighbor(localSubFaceIdx, 0) || scvf_.hasHalfParallelNeighbor(localSubFaceIdx) || scvf_.hasCornerParallelNeighbor(localSubFaceIdx))
         {
-            const Scalar boundaryVelocity = getParallelVelocityFromBoundary_(element_, scvf_, faceVars_, currentScvfBoundaryTypes, lateralFaceBoundaryTypes, localSubFaceIdx);
-            const Scalar boundaryMomentum = boundaryVelocity*insideDensity;
-            return {boundaryMomentum, boundaryMomentum, boundaryMomentum};
+            if ((scvf_.hasHalfParallelNeighbor(localSubFaceIdx) || scvf_.hasCornerParallelNeighbor(localSubFaceIdx)) && dirichletParallelNeighbor_(localSubFaceIdx))
+            {
+                const Scalar boundaryVelocity = getParallelVelocityFromCorner_(localSubFaceIdx);
+                const Scalar boundaryMomentum = boundaryVelocity*insideDensity;
+
+                return {boundaryMomentum, boundaryMomentum, boundaryMomentum};
+            }
+            else if (!scvf_.hasParallelNeighbor(localSubFaceIdx, 0))
+            {
+                const Scalar boundaryVelocity = getParallelVelocityFromBoundary_(element_, scvf_, faceVars_, currentScvfBoundaryTypes, lateralFaceBoundaryTypes, localSubFaceIdx);
+                const Scalar boundaryMomentum = boundaryVelocity*insideDensity;
+
+                return {boundaryMomentum, boundaryMomentum, boundaryMomentum};
+            }
         }
 
         if (selfIsUpstream)
@@ -327,7 +372,13 @@ private:
     {
         // If the lateral face lies on a boundary, we assume that the parallel velocity on the boundary is actually known,
         // thus we always use this value for the computation of the transported momentum.
-        if (!scvf_.hasParallelNeighbor(localSubFaceIdx, 0))
+        if ((scvf_.hasHalfParallelNeighbor(localSubFaceIdx) || scvf_.hasCornerParallelNeighbor(localSubFaceIdx)) && dirichletParallelNeighbor_(localSubFaceIdx))
+        {
+            const Scalar boundaryVelocity =  getParallelVelocityFromCorner_(localSubFaceIdx);
+            const Scalar boundaryMomentum = boundaryVelocity*outsideDensity;
+            return {boundaryMomentum, boundaryMomentum};
+        }
+        else if (!scvf_.hasParallelNeighbor(localSubFaceIdx, 0))
         {
             const Scalar boundaryVelocity = getParallelVelocityFromBoundary_(element_, scvf_, faceVars_, currentScvfBoundaryTypes, lateralFaceBoundaryTypes, localSubFaceIdx);
             const Scalar boundaryMomentum = boundaryVelocity*insideDensity;
@@ -450,6 +501,155 @@ private:
         return getParallelVelocityFromBoundary_(element, scvf, faceVars, currentScvfBoundaryTypes, lateralOppositeFaceBoundaryTypes, localOppositeSubFaceIdx);
     }
 
+    /*!
+     * \brief Returns the boundary subcontrolvolumeface in a corner geometry.
+     *
+     * ------------
+     * |     xxxx o
+     * |     xxxx o
+     * |     xxxx o
+     * ------------bbbbbbbbbbb
+     * |     yyyy o          |
+     * |     yyyy o          |
+     * |     yyyy o          |
+     * -----------------------
+     *
+     * This function will be entered in such a corner geometry (there is no cell in the upper right, --- and |
+     * stand for the grid cells). The scvf_ will be one of the two ones denoted by o (upper one
+     * hasCornerParallelNeighbor, lower one hasHalfParallelNeighbor). x and y are the two possible corresponding
+     * half-control volumes. In both cases, the returned boundaryScvf is the one marked by b. It needs to be the
+     * same boundaryScvf returned for the sake of flux continuity.
+     */
+    const SubControlVolumeFace& boundaryScvf_(const int localSubFaceIdx) const
+    {
+        if (scvf_.hasHalfParallelNeighbor(localSubFaceIdx))
+        {
+            return fvGeometry_.scvf(scvf_.outsideScvIdx(), scvf_.pairData(localSubFaceIdx).localLateralFaceIdx);
+        }
+        else if (scvf_.hasCornerParallelNeighbor(localSubFaceIdx))
+        {
+           /*
+            *      ------------
+            *      | xxxxxxxx o
+            *      | xxxxxxxx o
+            *      | xxxxxxxx o
+            *      lllllllllll-bbbbbbbbbbb
+            *      | yyyyyyyy p          |
+            *      | yyyyyyyy p          |
+            *      | yyyyyyyy p          |
+            *      -----------------------
+            *
+            * o: scvf_, l: lateralFace, p: parallelFace, b: returned scvf, x: scvf_ inside scv, y: lateralFace
+            * outside scv
+            */
+            const SubControlVolumeFace& lateralFace = fvGeometry_.scvf(scvf_.insideScvIdx(), scvf_.pairData(localSubFaceIdx).localLateralFaceIdx);
+            const SubControlVolumeFace& parallelFace = fvGeometry_.scvf(lateralFace.outsideScvIdx(), scvf_.localFaceIdx());
+
+            const auto& localLateralIdx = scvf_.pairData(localSubFaceIdx).localLateralFaceIdx;
+            const auto& localLateralOppositeIdx =  (localLateralIdx % 2) ? (localLateralIdx - 1) : (localLateralIdx + 1);
+
+            return fvGeometry_.scvf(parallelFace.outsideScvIdx(), localLateralOppositeIdx);
+        }
+        else
+        {
+            DUNE_THROW(Dune::InvalidStateException, "The function boundaryScvf_ should only be called when hasHalfParallelNeighbor or hasCornerParallelNeighbor.");
+        }
+    }
+
+   /*!
+    * \brief Gets the boundary element in a corner geometry.
+    *
+    * ------------
+    * |     xxxx o
+    * |     xxxx o
+    * |     xxxx o
+    * -----------------------
+    * |     yyyy o bbbbbbbb |
+    * |     yyyy o bbbbbbbb |
+    * |     yyyy o bbbbbbbb |
+    * -----------------------
+    *
+    * This function will be entered in such a corner geometry (there is no cell in the upper right, --- and |
+    * stand for the grid cells). The scvf_ will be one of the two ones denoted by o (upper one
+    * hasCornerParallelNeighbor, lower one hasHalfParallelNeighbor). x and y are the two possible corresponding
+    * half-control volumes. In both cases, the returned boundaryElement is the one marked by b.  It needs to be
+    * the same boundaryScvf returned for the sake of flux continuity.
+    */
+    Element boundaryElement_(const int localSubFaceIdx) const
+    {
+        if (scvf_.hasHalfParallelNeighbor(localSubFaceIdx))
+        {
+            return fvGeometry_.gridGeometry().element(scvf_.outsideScvIdx());
+        }
+        else if (scvf_.hasCornerParallelNeighbor(localSubFaceIdx))
+        {
+            const SubControlVolumeFace& lateralFace = fvGeometry_.scvf(scvf_.insideScvIdx(), scvf_.pairData(localSubFaceIdx).localLateralFaceIdx);
+            const SubControlVolumeFace& parallelFace = fvGeometry_.scvf(lateralFace.outsideScvIdx(), scvf_.localFaceIdx());
+
+            return fvGeometry_.gridGeometry().element(parallelFace.outsideScvIdx());
+        }
+        else
+        {
+            DUNE_THROW(Dune::InvalidStateException, "When entering boundaryElement_ scvf_ should have either hasHalfParallelNeighbor or hasCornerParallelNeighbor true. Not the case here.");
+        }
+    }
+
+   /*!
+    * \brief Sets the bools hasDirichletCornerParallelNeighbor and hasDirichletHalfParallelNeighbor.
+    *
+    * ------------
+    * |     xxxx o
+    * |     xxxx o
+    * |     xxxx o
+    * ------------bbbbbbbbbbb
+    * |     yyyy o          |
+    * |     yyyy o boundary |
+    * |     yyyy o  element |
+    * -----------------------
+    *
+    * This function will be entered in such a corner geometry (there is no cell in the upper right, --- and |
+    * stand for the grid cells). The scvf_ will be one of the two ones denoted by o (upper one
+    * hasCornerParallelNeighbor, lower one hasHalfParallelNeighbor). x and y are the two possible corresponding
+    * half-control volumes. In both cases, we check if the face bbb, part of the edge of element boundaryElement,
+    * is a Dirichlet boundary.
+    */
+    bool dirichletParallelNeighbor_(const int localSubFaceIdx) const
+    {
+        const auto& problem = elemVolVars_.gridVolVars().problem();
+        const Element& boundaryElement = boundaryElement_(localSubFaceIdx);
+        const SubControlVolumeFace& boundaryScvf = boundaryScvf_(localSubFaceIdx);
+
+        return problem.boundaryTypes(boundaryElement, boundaryScvf).isDirichlet(Indices::velocity(scvf_.directionIndex()));
+    }
+
+   /*!
+    * \brief Gets the parallel velocity from a corner geometry.
+    *
+    * ------------
+    * |     xxxx o
+    * |     xxxx o
+    * |     xxxx o
+    * -----------*-----------
+    * |     yyyy o          |
+    * |     yyyy o          |
+    * |     yyyy o          |
+    * -----------------------
+    *
+    * This function will be entered in such a corner geometry (there is no cell in the upper right, --- and |
+    * stand for the grid cells). The scvf_ will be one of the two ones denoted by o (upper one
+    * hasCornerParallelNeighbor, lower one hasHalfParallelNeighbor). x and y are the two possible corresponding
+    * half-control volumes. In both cases, the returned velocity is situated in the corner (*).
+    */
+    Scalar getParallelVelocityFromCorner_(const int localSubFaceIdx) const
+    {
+        const auto& problem = elemVolVars_.gridVolVars().problem();
+        const Element& boundaryElement = boundaryElement_(localSubFaceIdx);
+        const SubControlVolumeFace& boundaryScvf = boundaryScvf_(localSubFaceIdx);
+        const auto ghostFace = makeStaggeredBoundaryFace(boundaryScvf, scvf_.pairData(localSubFaceIdx).lateralStaggeredFaceCenter);
+
+        return problem.dirichlet(boundaryElement, ghostFace)[Indices::velocity(scvf_.directionIndex())];
+    }
+
     const Element& element_;
     const FVElementGeometry& fvGeometry_;
     const SubControlVolumeFace& scvf_;
diff --git a/dumux/freeflow/navierstokes/staggered/velocitygradients.hh b/dumux/freeflow/navierstokes/staggered/velocitygradients.hh
index b2c9d33a1cd97925efb797a6ad9c9f69d02ad781..932e1687ac51f5c570d811791e917ff63b539c11 100644
--- a/dumux/freeflow/navierstokes/staggered/velocitygradients.hh
+++ b/dumux/freeflow/navierstokes/staggered/velocitygradients.hh
@@ -93,6 +93,26 @@ public:
      *
      *                                                       O position at which gradient is evaluated
      * \endverbatim
+     *
+     * ------------
+     * |     xxxx s
+     * |     xxxx a
+     * |     xxxx s
+     * -----------O-----------
+     * |     yyyy s zzzz     |
+     * |     yyyy b zzzz     |
+     * |     yyyy s zzzz     |
+     * -----------------------
+     *
+     * In a corner geometry (scvf is sas or sbs), we calculate the velocity gradient at O, by
+     * (velocity(a)-velocity(b))/distance(a,b) for the half-control volumes x and y, but by
+     * (velocity(O)-velocity(b))/distance(O,b) for z. This does not harm flux continuity (x and y use the same
+     * formulation). We do this different formulation for y (approximate gradient by central differncing) and
+     * z (approximate gradient by forward/backward differencing), because it is the natural way of implementing
+     * it and it is not clear which gradient is the better approximation in this case anyway.
+     * Particularly, for the frequent case of no-slip, no-flow boundaries, the velocity would be zero at O and a
+     * and thus, the gradient within the flow domain might be better approximated by velocity(b)/distanc(O,b)
+     * than by velocity(b)/distance(a,b).
      */
     template<class Problem, class FaceVariables>
     static Scalar velocityGradIJ(const Problem& problem,
diff --git a/test/freeflow/navierstokes/channel/3d/CMakeLists.txt b/test/freeflow/navierstokes/channel/3d/CMakeLists.txt
index 3e572be57131d5e241d545ff698305460ec2ba0f..11356d3181361dc964ea2ff5cdb069fedf48ba23 100644
--- a/test/freeflow/navierstokes/channel/3d/CMakeLists.txt
+++ b/test/freeflow/navierstokes/channel/3d/CMakeLists.txt
@@ -1,22 +1,34 @@
-dumux_add_test(NAME test_ff_stokes_channel_3d
+add_executable(test_ff_stokes_channel_3d EXCLUDE_FROM_ALL main.cc)
+target_compile_definitions(test_ff_stokes_channel_3d PUBLIC "GRID_DIM=3")
+
+dumux_add_test(NAME test_ff_stokes_channel_3d_cuboid
+              TARGET test_ff_stokes_channel_3d
               LABELS freeflow navierstokes
-              SOURCES main.cc
-              LABELS freeflow
-              COMPILE_DEFINITIONS DIM_3D=1
               CMAKE_GUARD HAVE_UMFPACK
               COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
               CMD_ARGS       --script fuzzy
-                             --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_stokes_channel_3d-reference.vtu
-                                     ${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel_3d-00001.vtu
+                             --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_stokes_channel_3d_cuboid-reference.vtu
+                                     ${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel_3d_cuboid-00001.vtu
+                             --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel_3d params.input
+                             -Problem.Name test_ff_stokes_channel_3d_cuboid"
+                             --zeroThreshold {"velocity_liq \(m/s\)":1e-12})
+
+dumux_add_test(NAME test_ff_stokes_channel_3d_staircase
+              TARGET test_ff_stokes_channel_3d
+              LABELS freeflow navierstokes
+              CMAKE_GUARD HAVE_UMFPACK dune-subgrid_FOUND
+              COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+              CMD_ARGS       --script fuzzy
+                             --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_stokes_channel_3d_staircase-reference.vtu
+                                     ${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel_3d_staircase-00001.vtu
                              --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_stokes_channel_3d params.input
-                             -Problem.Name test_ff_stokes_channel_3d"
+                             -Problem.Name test_ff_stokes_channel_3d_staircase -Problem.IsStaircaseGeometry true"
                              --zeroThreshold {"velocity_liq \(m/s\)":1e-12})
 
 dumux_add_test(NAME test_ff_stokes_channel_pseudo3d
               LABELS freeflow navierstokes
               SOURCES main.cc
-              LABELS freeflow
-              COMPILE_DEFINITIONS DIM_3D=0
+              COMPILE_DEFINITIONS GRID_DIM=2
               CMAKE_GUARD HAVE_UMFPACK
               COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
               CMD_ARGS       --script fuzzy
diff --git a/test/freeflow/navierstokes/channel/3d/main.cc b/test/freeflow/navierstokes/channel/3d/main.cc
index ee1987a48afd96a92441c19ffd84fcbd7dfbea15..3e77d0573fcfd3106ca8a34c8fc74b01455a1ce6 100644
--- a/test/freeflow/navierstokes/channel/3d/main.cc
+++ b/test/freeflow/navierstokes/channel/3d/main.cc
@@ -39,7 +39,8 @@
 #include <dumux/common/parameters.hh>
 #include <dumux/common/properties.hh>
 #include <dumux/freeflow/navierstokes/staggered/fluxoversurface.hh>
-#include <dumux/io/grid/gridmanager.hh>
+#include <dumux/io/grid/gridmanager_sub.hh>
+#include <dumux/io/grid/gridmanager_yasp.hh>
 #include <dumux/io/staggeredvtkoutputmodule.hh>
 #include <dumux/linear/seqsolverbackend.hh>
 #include <dumux/nonlinear/newtonsolver.hh>
@@ -63,9 +64,33 @@ int main(int argc, char** argv) try
     // parse command line arguments and input file
     Parameters::init(argc, argv);
 
-    // try to create a grid (from the given grid file or the input file)
-    GridManager<GetPropType<TypeTag, Properties::Grid>> gridManager;
+    // create a grid
+    using Grid = GetPropType<TypeTag, Properties::Grid>;
+    Dumux::GridManager<Grid> gridManager;
+
+#if HAVE_DUNE_SUBGRID
+    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
+    const bool isStaircaseGeometry = getParam<bool>("Problem.IsStaircaseGeometry", false);
+
+    auto selector = [&](const auto& element)
+    {
+        if (!isStaircaseGeometry)
+            return true;
+
+        const Scalar deltaX = 0.003;
+        const Scalar deltaZ = 0.000075;
+        const Scalar deltaY = 0.0003;
+
+        const Scalar eps = 1e-8;
+        const auto globalPos = element.geometry().center();
+
+        return globalPos[2] > (deltaZ/deltaX * globalPos[0] + deltaZ/deltaY * globalPos[1] - deltaZ + eps);
+    };
+
+    gridManager.init(selector, "Internal");
+#else
     gridManager.init();
+#endif
 
     ////////////////////////////////////////////////////////////
     // run instationary non-linear problem on this grid
@@ -118,14 +143,14 @@ int main(int argc, char** argv) try
                     GetPropType<TypeTag, Properties::ModelTraits>,
                     GetPropType<TypeTag, Properties::LocalResidual>> flux(*gridVariables, x);
     using GridView = typename GetPropType<TypeTag, Properties::GridGeometry>::GridView;
-    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
     using GlobalPosition = Dune::FieldVector<Scalar, GridView::dimensionworld>;
 
     const Scalar xMin = gridGeometry->bBoxMin()[0];
     const Scalar xMax = gridGeometry->bBoxMax()[0];
     const Scalar yMin = gridGeometry->bBoxMin()[1];
     const Scalar yMax = gridGeometry->bBoxMax()[1];
-#if DIM_3D
+
+#if GRID_DIM == 3
     const Scalar zMin = gridGeometry->bBoxMin()[2];
     const Scalar zMax = gridGeometry->bBoxMax()[2];
 #endif
@@ -136,7 +161,7 @@ int main(int argc, char** argv) try
     // In this case, we add half a cell-width to the x-position in order to make sure that
     // the cell faces lie on the plane. This assumes a regular cartesian grid.
     // The second plane is placed at the outlet of the channel.
-#if DIM_3D
+#if GRID_DIM == 3
     const auto p0inlet = GlobalPosition{xMin, yMin, zMin};
     const auto p1inlet = GlobalPosition{xMin, yMax, zMin};
     const auto p2inlet = GlobalPosition{xMin, yMin, zMax};
@@ -157,7 +182,7 @@ int main(int argc, char** argv) try
 
     const Scalar offsetX = (numCellsX % 2 == 0) ? 0.0 : 0.5*((xMax - xMin) / numCellsX);
 
-#if DIM_3D
+#if GRID_DIM == 3
     const auto p0middle = GlobalPosition{planePosMiddleX + offsetX, yMin, zMin};
     const auto p1middle = GlobalPosition{planePosMiddleX + offsetX, yMax, zMin};
     const auto p2middle = GlobalPosition{planePosMiddleX + offsetX, yMin, zMax};
@@ -170,7 +195,7 @@ flux.addSurface("middle", p0middle, p1middle);
 #endif
 
     // The second plane is placed at the outlet of the channel.
-#if DIM_3D
+#if GRID_DIM == 3
     const auto p0outlet = GlobalPosition{xMax, yMin, zMin};
     const auto p1outlet = GlobalPosition{xMax, yMax, zMin};
     const auto p2outlet = GlobalPosition{xMax, yMin, zMax};
diff --git a/test/freeflow/navierstokes/channel/3d/problem.hh b/test/freeflow/navierstokes/channel/3d/problem.hh
index 243804b332693c46223d692a0bf585f2af99cbb8..5d95ea95b0152010a49e886137636b049ad86b3a 100644
--- a/test/freeflow/navierstokes/channel/3d/problem.hh
+++ b/test/freeflow/navierstokes/channel/3d/problem.hh
@@ -31,6 +31,10 @@
 #include <dune/common/float_cmp.hh>
 #include <dune/grid/yaspgrid.hh>
 
+#if HAVE_DUNE_SUBGRID
+#include <dune/subgrid/subgrid.hh>
+#endif
+
 #include <dumux/discretization/staggered/freeflow/properties.hh>
 
 #include <dumux/freeflow/navierstokes/boundarytypes.hh>
@@ -40,10 +44,6 @@
 #include <dumux/material/components/constant.hh>
 #include <dumux/material/fluidsystems/1pliquid.hh>
 
-#ifndef DIM_3D
-#define DIM_3D 0
-#endif
-
 namespace Dumux {
 
 template <class TypeTag>
@@ -64,13 +64,19 @@ struct FluidSystem<TypeTag, TTag::ThreeDChannelTest>
 };
 
 // Set the grid type
-#if DIM_3D
 template<class TypeTag>
-struct Grid<TypeTag, TTag::ThreeDChannelTest> { using type = Dune::YaspGrid<3>; };
+struct Grid<TypeTag, TTag::ThreeDChannelTest>
+{
+    static constexpr int dim = GRID_DIM;
+
+    using HostGrid = Dune::YaspGrid<dim, Dune::EquidistantOffsetCoordinates<GetPropType<TypeTag, Properties::Scalar>, dim> >;
+
+#if HAVE_DUNE_SUBGRID
+    using type = Dune::SubGrid<HostGrid::dimension, HostGrid>;
 #else
-template<class TypeTag>
-struct Grid<TypeTag, TTag::ThreeDChannelTest> { using type = Dune::YaspGrid<2>; };
+    using type = HostGrid;
 #endif
+};
 
 // Set the problem property
 template<class TypeTag>
@@ -122,7 +128,7 @@ class ThreeDChannelTestProblem : public NavierStokesProblem<TypeTag>
     using PrimaryVariables = GetPropType<TypeTag, Properties::PrimaryVariables>;
     using NumEqVector = GetPropType<TypeTag, Properties::NumEqVector>;
 
-    static constexpr bool enablePseudoThreeDWallFriction = !DIM_3D;
+    static constexpr bool enablePseudoThreeDWallFriction = !(GRID_DIM == 3);
 
 public:
     ThreeDChannelTestProblem(std::shared_ptr<const GridGeometry> gridGeometry)
@@ -170,7 +176,7 @@ public:
     {
         auto source = NumEqVector(0.0);
 
-#if !DIM_3D
+#if GRID_DIM != 3
             static const Scalar height = getParam<Scalar>("Problem.Height");
             static const Scalar factor = getParam<Scalar>("Problem.PseudoWallFractionFactor", 8.0);
             source[scvf.directionIndex()] = this->pseudo3DWallFriction(scvf, elemVolVars, elemFaceVars, height, factor);
diff --git a/test/freeflow/navierstokes/kovasznay/CMakeLists.txt b/test/freeflow/navierstokes/kovasznay/CMakeLists.txt
index e5bbb76d93f7dcfc1f318faa73915c6f121b6a7d..1ef2995ed0afe33e570671ec274f0899f8e09832 100644
--- a/test/freeflow/navierstokes/kovasznay/CMakeLists.txt
+++ b/test/freeflow/navierstokes/kovasznay/CMakeLists.txt
@@ -10,17 +10,33 @@ dumux_add_test(NAME test_ff_navierstokes_kovasznay
                              --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_navierstokes_kovasznay params.input
                              -Problem.Name test_ff_navierstokes_kovasznay")
 
-dumux_add_test(NAME test_ff_navierstokes_kovasznay_higherorder
-              SOURCES main.cc
+add_executable(test_ff_navierstokes_kovasznay_higherorder EXCLUDE_FROM_ALL main.cc)
+target_compile_definitions(test_ff_navierstokes_kovasznay_higherorder PUBLIC "UPWINDSCHEMEORDER=2")
+
+dumux_add_test(NAME test_ff_navierstokes_kovasznay_higherorder_cuboid
+              TARGET test_ff_navierstokes_kovasznay_higherorder
               LABELS freeflow navierstokes
               CMAKE_GUARD HAVE_UMFPACK
-              COMPILE_DEFINITIONS UPWINDSCHEMEORDER=2
               COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
               CMD_ARGS       --script fuzzy
-                             --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_navierstokes_kovasznay_higherorder-reference.vtu
-                                     ${CMAKE_CURRENT_BINARY_DIR}/test_ff_navierstokes_kovasznay_higherorder-00001.vtu
+                             --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_navierstokes_kovasznay_higherorder_cuboid-reference.vtu
+                                     ${CMAKE_CURRENT_BINARY_DIR}/test_ff_navierstokes_kovasznay_higherorder_cuboid-00001.vtu
+                             --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_navierstokes_kovasznay_higherorder params.input
+                             -Problem.Name test_ff_navierstokes_kovasznay_higherorder_cuboid
+                             -Flux.TvdApproach Hou
+                             -Flux.DifferencingScheme Vanleer")
+
+dumux_add_test(NAME test_ff_navierstokes_kovasznay_higherorder_staircase
+              TARGET test_ff_navierstokes_kovasznay_higherorder
+              LABELS freeflow navierstokes
+              CMAKE_GUARD HAVE_UMFPACK dune-subgrid_FOUND
+              COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+              CMD_ARGS       --script fuzzy
+                             --files ${CMAKE_SOURCE_DIR}/test/references/test_ff_navierstokes_kovasznay_higherorder_staircase-reference.vtu
+                                     ${CMAKE_CURRENT_BINARY_DIR}/test_ff_navierstokes_kovasznay_higherorder_staircase-00001.vtu
                              --command "${CMAKE_CURRENT_BINARY_DIR}/test_ff_navierstokes_kovasznay_higherorder params.input
-                             -Problem.Name test_ff_navierstokes_kovasznay_higherorder
+                             -Problem.Name test_ff_navierstokes_kovasznay_higherorder_staircase
+                             -Problem.IsStaircaseGeometry true
                              -Flux.TvdApproach Hou
                              -Flux.DifferencingScheme Vanleer")
 
diff --git a/test/freeflow/navierstokes/kovasznay/main.cc b/test/freeflow/navierstokes/kovasznay/main.cc
index 993cc579be2a362f97dc8a40e0b90c6f23156e3c..29ee863b0907cfd87b1a76909220474273b254d2 100644
--- a/test/freeflow/navierstokes/kovasznay/main.cc
+++ b/test/freeflow/navierstokes/kovasznay/main.cc
@@ -38,6 +38,7 @@
 #include <dumux/common/dumuxmessage.hh>
 #include <dumux/common/parameters.hh>
 #include <dumux/common/properties.hh>
+#include <dumux/io/grid/gridmanager_sub.hh>
 #include <dumux/io/grid/gridmanager_yasp.hh>
 #include <dumux/io/staggeredvtkoutputmodule.hh>
 #include <dumux/linear/seqsolverbackend.hh>
@@ -63,9 +64,37 @@ int main(int argc, char** argv) try
     // parse command line arguments and input file
     Parameters::init(argc, argv);
 
-    // try to create a grid (from the given grid file or the input file)
-    GridManager<GetPropType<TypeTag, Properties::Grid>> gridManager;
+    // create a grid
+    using Grid = GetPropType<TypeTag, Properties::Grid>;
+    Dumux::GridManager<Grid> gridManager;
+
+#if HAVE_DUNE_SUBGRID
+    const bool isStaircaseGeometry = getParam<bool>("Problem.IsStaircaseGeometry", false);
+
+    // cut out elements within the stair-case region
+    auto selector = [&](const auto& element)
+    {
+        if (!isStaircaseGeometry)
+            return true;
+
+        const auto globalPos = element.geometry().center();
+        const auto x = globalPos[0];
+        const auto y = globalPos[1];
+
+        auto eps = 1e-12;
+        static const auto lowerLeft = getParam<std::decay_t<decltype(globalPos)>>("Grid.LowerLeft");
+
+        if (x < eps || y > (lowerLeft[1] + 1.0 - eps))
+            return true;
+
+        const bool isBelowCurve = y < (x/2.0 - std::floor(x/2.0) + lowerLeft[1] + eps);
+        return !isBelowCurve;
+    };
+
+    gridManager.init(selector, "Internal");
+#else
     gridManager.init();
+#endif
 
     ////////////////////////////////////////////////////////////
     // run instationary non-linear problem on this grid
diff --git a/test/freeflow/navierstokes/kovasznay/problem.hh b/test/freeflow/navierstokes/kovasznay/problem.hh
index cf632fde7f0e969404424279e3ffb916e9ec4f9b..f40ff80119e2d805cd3e4380ede6afdc28d27753 100644
--- a/test/freeflow/navierstokes/kovasznay/problem.hh
+++ b/test/freeflow/navierstokes/kovasznay/problem.hh
@@ -31,6 +31,10 @@
 
 #include <dune/grid/yaspgrid.hh>
 
+#if HAVE_DUNE_SUBGRID
+#include <dune/subgrid/subgrid.hh>
+#endif
+
 #include <dumux/discretization/staggered/freeflow/properties.hh>
 
 #include <dumux/freeflow/navierstokes/boundarytypes.hh>
@@ -62,7 +66,16 @@ struct FluidSystem<TypeTag, TTag::KovasznayTest>
 
 // Set the grid type
 template<class TypeTag>
-struct Grid<TypeTag, TTag::KovasznayTest> { using type = Dune::YaspGrid<2, Dune::EquidistantOffsetCoordinates<GetPropType<TypeTag, Properties::Scalar>, 2> >; };
+struct Grid<TypeTag, TTag::KovasznayTest>
+{
+    using HostGrid = Dune::YaspGrid<2, Dune::EquidistantOffsetCoordinates<GetPropType<TypeTag, Properties::Scalar>, 2> >;
+
+#if HAVE_DUNE_SUBGRID
+    using type = Dune::SubGrid<HostGrid::dimension, HostGrid>;
+#else
+    using type = HostGrid;
+#endif
+};
 
 // Set the problem property
 template<class TypeTag>
diff --git a/test/references/test_ff_navierstokes_kovasznay_higherorder-reference.vtu b/test/references/test_ff_navierstokes_kovasznay_higherorder_cuboid-reference.vtu
similarity index 100%
rename from test/references/test_ff_navierstokes_kovasznay_higherorder-reference.vtu
rename to test/references/test_ff_navierstokes_kovasznay_higherorder_cuboid-reference.vtu
diff --git a/test/references/test_ff_navierstokes_kovasznay_higherorder_staircase-reference.vtu b/test/references/test_ff_navierstokes_kovasznay_higherorder_staircase-reference.vtu
new file mode 100644
index 0000000000000000000000000000000000000000..1ae575dcb715b48915fd7c2c8e58ad61e4b53374
--- /dev/null
+++ b/test/references/test_ff_navierstokes_kovasznay_higherorder_staircase-reference.vtu
@@ -0,0 +1,3228 @@
+<?xml version="1.0"?>
+<VTKFile type="UnstructuredGrid" version="0.1" byte_order="LittleEndian">
+  <UnstructuredGrid>
+    <Piece NumberOfCells="2000" NumberOfPoints="2101">
+      <CellData Scalars="p" Vectors="velocity_liq (m/s)">
+        <DataArray type="Float32" Name="p" NumberOfComponents="1" format="ascii">
+          -0.749079 -0.639201 -0.529853 -0.433893 -0.347364 -0.269033 -0.197946 -0.133323 -0.074556 -0.0215656 0.0424444 -0.749079
+          -0.637301 -0.530172 -0.434434 -0.347894 -0.269491 -0.198324 -0.133595 -0.0745197 -0.019938 0.0321599 0.0762326 -0.749079
+          -0.635321 -0.529766 -0.434471 -0.348125 -0.269809 -0.198688 -0.134004 -0.0750519 -0.0211883 0.0287192 0.072998 0.102088
+          0.15187 -0.749079 -0.633852 -0.529178 -0.434252 -0.348137 -0.269974 -0.198973 -0.134419 -0.075673 -0.0221984 0.0264924
+          0.0697118 0.107075 0.147736 0.173518 0.212367 -0.749079 -0.632789 -0.528613 -0.433943 -0.348025 -0.270016 -0.199149
+          -0.134736 -0.0761685 -0.0229339 0.025387 0.0689063 0.108088 0.145093 0.176661 0.209582 0.236289 -0.749079 -0.631938
+          -0.528074 -0.433579 -0.347814 -0.269935 -0.199184 -0.134886 -0.0764436 -0.0233359 0.0248876 0.0685892 0.108235 0.144613
+          0.177325 0.208061 0.235701 0.257006 0.282238 -0.749079 -0.631426 -0.52771 -0.433324 -0.347662 -0.269872 -0.199202
+          -0.134982 -0.0766171 -0.0235774 0.0246114 0.0683778 0.108154 0.144404 0.177317 0.207444 0.234511 0.258546 0.28217
+          0.301364 -0.749079 -0.632308 -0.528379 -0.433997 -0.348324 -0.270511 -0.199812 -0.135566 -0.077175 -0.0241044 0.0241298
+          0.0679632 0.107801 0.144044 0.176977 0.206971 0.234122 0.258622 0.281242 0.301717 0.319326 0.334859 -0.749079
+          -0.631905 -0.527943 -0.433652 -0.348046 -0.27028 -0.199618 -0.135403 -0.0770399 -0.0239933 0.0242227 0.0680476 0.107883
+          0.144099 0.177018 0.206946 0.234106 0.258736 0.281154 0.301331 0.319425 0.336234 0.351762 0.363108 -0.749079
+          -0.631551 -0.527707 -0.433512 -0.347953 -0.270207 -0.199554 -0.135343 -0.0769838 -0.0239399 0.0242738 0.0680968 0.107929
+          0.144133 0.177037 0.206938 0.23409 0.258732 0.281096 0.301319 0.319577 0.336176 0.351346 0.364855 0.375969
+          -0.749079 -0.630964 -0.52747 -0.433406 -0.347895 -0.270166 -0.199516 -0.135302 -0.0769386 -0.0238909 0.0243251 0.0681488
+          0.107979 0.144176 0.17707 0.206954 0.234093 0.258726 0.28107 0.301309 0.319628 0.336247 0.351344 0.364956
+          0.37745 0.389599 0.396408 -0.749079 -0.630433 -0.527269 -0.433322 -0.347854 -0.270138 -0.19949 -0.135272 -0.0769017
+          -0.0238481 0.0243725 0.0681986 0.108028 0.144223 0.177109 0.206982 0.234109 0.258729 0.281062 0.301304 0.319645
+          0.336278 0.351368 0.365046 0.37751 0.388966 0.398658 0.406783 -0.749079 -0.630217 -0.527184 -0.433289 -0.347839
+          -0.270127 -0.199476 -0.135252 -0.0768758 -0.0238159 0.0244097 0.0682391 0.10807 0.144262 0.177143 0.207009 0.234126
+          0.258736 0.281061 0.301303 0.319653 0.336294 0.351392 0.365094 0.377553 0.388877 0.39902 0.408493 0.418317
+          0.422266 -0.749079 -0.630479 -0.527327 -0.433385 -0.347914 -0.270189 -0.199525 -0.13529 -0.0769031 -0.0238335 0.0244002
+          0.0682357 0.10807 0.144264 0.177144 0.207005 0.234117 0.258722 0.281044 0.301287 0.319644 0.336293 0.351401
+          0.365117 0.377584 0.388918 0.399194 0.408699 0.417519 0.424625 0.433289 0.4363 -0.749079 -0.631061 -0.527587
+          -0.433529 -0.34801 -0.27026 -0.199581 -0.135334 -0.0769361 -0.023858 0.0243825 0.0682229 0.10806 0.144255 0.177133
+          0.206992 0.234099 0.258701 0.281022 0.301268 0.319631 0.33629 0.351408 0.365137 0.377614 0.38896 0.399287
+          0.408754 0.417401 0.425072 0.432512 0.438206 0.443485 -0.749079 -0.631705 -0.527879 -0.433688 -0.348112 -0.270333
+          -0.199637 -0.135378 -0.0769717 -0.0238867 0.0243587 0.0682023 0.108041 0.144235 0.177112 0.206968 0.234073 0.258673
+          0.280995 0.301245 0.319615 0.336284 0.351414 0.365155 0.377644 0.389003 0.399348 0.408794 0.417399 0.425185
+          0.432363 0.438621 0.444698 0.45153 0.453563 -0.749079 -0.632111 -0.528158 -0.433863 -0.348233 -0.270425 -0.19971
+          -0.135438 -0.077022 -0.0239303 0.0243195 0.0681655 0.108005 0.144198 0.177073 0.206927 0.234031 0.258632 0.280957
+          0.301213 0.319593 0.336273 0.351416 0.36517 0.377671 0.389042 0.399396 0.408834 0.417429 0.425243 0.432379
+          0.438834 0.444911 0.450668 0.454883 0.459179 -0.749079 -0.632534 -0.528597 -0.43421 -0.348513 -0.270658 -0.199911
+          -0.135614 -0.0771796 -0.0240732 0.0241881 0.0680428 0.107889 0.144087 0.176966 0.206824 0.233933 0.25854 0.280873
+          0.301141 0.319534 0.33623 0.351389 0.365159 0.377675 0.389059 0.399422 0.40886 0.417455 0.425282 0.432419
+          0.438928 0.444942 0.450445 0.455166 0.45975 0.464887 0.466956 -0.749079 -0.631613 -0.527904 -0.433529 -0.347856
+          -0.270036 -0.199329 -0.135076 -0.0766845 -0.0236213 0.0245978 0.0684118 0.108219 0.144381 0.177227 0.207055 0.234138
+          0.258724 0.281041 0.301297 0.319683 0.336376 0.351534 0.365305 0.377822 0.389206 0.399565 0.408998 0.417587
+          0.425411 0.432545 0.439058 0.445026 0.450461 0.455343 0.459925 0.464283 0.46753 0.471492 0.473787 -0.749079
+          -0.6319 -0.528101 -0.433669 -0.347947 -0.270095 -0.199368 -0.135105 -0.0767097 -0.0236466 0.0245699 0.06838 0.108183
+          0.144341 0.177183 0.20701 0.234093 0.258683 0.281006 0.301271 0.319669 0.336373 0.351545 0.365328 0.377857
+          0.38925 0.399615 0.409051 0.417642 0.425467 0.4326 0.439109 0.445053 0.450469 0.45539 0.459926 0.46408
+          0.467662 0.471124 0.473877 0.477045 -0.749079 -0.632291 -0.528365 -0.433879 -0.348108 -0.270218 -0.199465 -0.135185
+          -0.0767806 -0.0237128 0.0245054 0.0683153 0.108117 0.144274 0.177116 0.206945 0.234032 0.258627 0.280959 0.301234
+          0.319645 0.336363 0.351548 0.365345 0.377885 0.389288 0.399661 0.409101 0.417694 0.42552 0.432651 0.439152
+          0.445081 0.450484 0.455404 0.459902 0.46399 0.467644 0.470994 0.473935 0.476708 0.479167 0.482118 -0.749079
+          -0.63252 -0.528498 -0.433988 -0.348191 -0.270279 -0.199512 -0.135223 -0.0768154 -0.0237477 0.0244684 0.0682753 0.108074
+          0.144229 0.177071 0.2069 0.23399 0.258591 0.280931 0.301216 0.319638 0.336368 0.351566 0.365375 0.377926
+          0.389337 0.399715 0.409158 0.417752 0.425577 0.432704 0.439197 0.445113 0.450503 0.45541 0.459881 0.46394
+          0.467605 0.47093 0.473921 0.476665 0.47916 0.48146 0.48409 -0.749079 -0.632605 -0.528543 -0.434046 -0.348246
+          -0.270327 -0.199554 -0.135262 -0.0768535 -0.0237867 0.0244275 0.0682322 0.108029 0.144184 0.177026 0.206858 0.233953
+          0.258561 0.280909 0.301205 0.319638 0.336381 0.35159 0.36541 0.37797 0.389389 0.399772 0.409218 0.417812
+          0.425634 0.432755 0.43924 0.445145 0.450521 0.455414 0.459865 0.463907 0.467567 0.470884 0.473886 0.476619
+          0.479104 0.481365 0.4834 0.484597 0.488036 -0.749079 -0.632485 -0.528527 -0.434083 -0.348301 -0.270386 -0.199612
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diff --git a/test/references/test_ff_stokes_channel_3d-reference.vtu b/test/references/test_ff_stokes_channel_3d_cuboid-reference.vtu
similarity index 100%
rename from test/references/test_ff_stokes_channel_3d-reference.vtu
rename to test/references/test_ff_stokes_channel_3d_cuboid-reference.vtu
diff --git a/test/references/test_ff_stokes_channel_3d_staircase-reference.vtu b/test/references/test_ff_stokes_channel_3d_staircase-reference.vtu
new file mode 100644
index 0000000000000000000000000000000000000000..efa2a3d02dec70f038922fc820fb3d7ef3b04c8c
--- /dev/null
+++ b/test/references/test_ff_stokes_channel_3d_staircase-reference.vtu
@@ -0,0 +1,334 @@
+<?xml version="1.0"?>
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