diff --git a/test/porousmediumflow/1p/implicit/CMakeLists.txt b/test/porousmediumflow/1p/implicit/CMakeLists.txt
index dbbc593c5a7ec9dcaf23f8f7dcd3a49b17e28ec7..567e61ab85d45ad7cc516304811afd89aca30907 100644
--- a/test/porousmediumflow/1p/implicit/CMakeLists.txt
+++ b/test/porousmediumflow/1p/implicit/CMakeLists.txt
@@ -1,5 +1,6 @@
add_subdirectory(pointsources)
add_subdirectory(incompressible)
+add_subdirectory(internaldirichlet)
add_subdirectory(compressible)
add_subdirectory(periodicbc)
add_subdirectory(fracture2d3d)
diff --git a/test/porousmediumflow/1p/implicit/incompressible/main.cc b/test/porousmediumflow/1p/implicit/incompressible/main.cc
index e2ea13669ef31d80699aebc1438853f83d8a8a64..de2f1624dc0c3bfc3da3407773ce67ab0e58c2c9 100644
--- a/test/porousmediumflow/1p/implicit/incompressible/main.cc
+++ b/test/porousmediumflow/1p/implicit/incompressible/main.cc
@@ -55,6 +55,7 @@
#include <dumux/assembly/fvassembler.hh>
#include "problem.hh"
+#include "../internaldirichlet/problem.hh"
//! Function to write out the scv-wise velocities (overload for mpfa)
template<class FVGridGeometry, class GridVariables, class Sol,
diff --git a/test/porousmediumflow/1p/implicit/internaldirichlet/CMakeLists.txt b/test/porousmediumflow/1p/implicit/internaldirichlet/CMakeLists.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4ec99b479b7301168ad8f0beee51ec65ade3b548
--- /dev/null
+++ b/test/porousmediumflow/1p/implicit/internaldirichlet/CMakeLists.txt
@@ -0,0 +1,21 @@
+dune_add_test(NAME test_1p_internaldirichlet_tpfa
+ SOURCES ${CMAKE_SOURCE_DIR}/test/porousmediumflow/1p/implicit/incompressible/main.cc
+ LABELS porousmediumflow 1p
+ COMPILE_DEFINITIONS TYPETAG=OnePInternalDirichletTpfa
+ COMPILE_DEFINITIONS NUMDIFFMETHOD=DiffMethod::analytic
+ COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+ CMD_ARGS --script fuzzy
+ --files ${CMAKE_SOURCE_DIR}/test/references/test_1p_internaldirichlet_tpfa-reference.vtu
+ ${CMAKE_CURRENT_BINARY_DIR}/test_1p_internaldirichlet_tpfa-00001.vtu
+ --command "${CMAKE_CURRENT_BINARY_DIR}/test_1p_internaldirichlet_tpfa ${CMAKE_SOURCE_DIR}/test/porousmediumflow/1p/implicit/incompressible/params.input -Problem.Name test_1p_internaldirichlet_tpfa -Problem.EnableGravity false")
+
+dune_add_test(NAME test_1p_internaldirichlet_box
+ SOURCES ${CMAKE_SOURCE_DIR}/test/porousmediumflow/1p/implicit/incompressible/main.cc
+ LABELS porousmediumflow 1p
+ COMPILE_DEFINITIONS TYPETAG=OnePInternalDirichletBox
+ COMPILE_DEFINITIONS NUMDIFFMETHOD=DiffMethod::analytic
+ COMMAND ${CMAKE_SOURCE_DIR}/bin/testing/runtest.py
+ CMD_ARGS --script fuzzy
+ --files ${CMAKE_SOURCE_DIR}/test/references/test_1p_internaldirichlet_box-reference.vtu
+ ${CMAKE_CURRENT_BINARY_DIR}/test_1p_internaldirichlet_box-00001.vtu
+ --command "${CMAKE_CURRENT_BINARY_DIR}/test_1p_internaldirichlet_box ${CMAKE_SOURCE_DIR}/test/porousmediumflow/1p/implicit/incompressible/params.input -Problem.Name test_1p_internaldirichlet_box -Problem.EnableGravity false")
diff --git a/test/porousmediumflow/1p/implicit/internaldirichlet/problem.hh b/test/porousmediumflow/1p/implicit/internaldirichlet/problem.hh
new file mode 100644
index 0000000000000000000000000000000000000000..a20fb77d91c65051459183cb8399f2519bc5a3c4
--- /dev/null
+++ b/test/porousmediumflow/1p/implicit/internaldirichlet/problem.hh
@@ -0,0 +1,129 @@
+// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
+// vi: set et ts=4 sw=4 sts=4:
+/*****************************************************************************
+ * See the file COPYING for full copying permissions. *
+ * *
+ * This program is free software: you can redistribute it and/or modify *
+ * it under the terms of the GNU General Public License as published by *
+ * the Free Software Foundation, either version 3 of the License, or *
+ * (at your option) any later version. *
+ * *
+ * This program is distributed in the hope that it will be useful, *
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of *
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
+ * GNU General Public License for more details. *
+ * *
+ * You should have received a copy of the GNU General Public License *
+ * along with this program. If not, see <http://www.gnu.org/licenses/>. *
+ *****************************************************************************/
+/*!
+ * \file
+ * \ingroup OnePTests
+ * \brief The properties for the incompressible test
+ */
+
+#ifndef DUMUX_INCOMPRESSIBLE_ONEP_TEST_PROBLEM_INTERNAL_DIRICHLET_HH
+#define DUMUX_INCOMPRESSIBLE_ONEP_TEST_PROBLEM_INTERNAL_DIRICHLET_HH
+
+#include <test/porousmediumflow/1p/implicit/incompressible/problem.hh>
+
+namespace Dumux {
+// forward declarations
+template<class TypeTag> class OnePTestProblemInternalDirichlet;
+
+namespace Properties {
+// create the type tag nodes
+// Create new type tags
+namespace TTag {
+struct OnePInternalDirichlet {};
+struct OnePInternalDirichletTpfa { using InheritsFrom = std::tuple<OnePInternalDirichlet, OnePIncompressibleTpfa>; };
+struct OnePInternalDirichletBox { using InheritsFrom = std::tuple<OnePInternalDirichlet, OnePIncompressibleBox>; };
+} // end namespace TTag
+
+// Set the problem type
+template<class TypeTag>
+struct Problem<TypeTag, TTag::OnePInternalDirichlet>
+{ using type = OnePTestProblemInternalDirichlet<TypeTag>; };
+
+} // end namespace Properties
+
+/*!
+ * \ingroup OnePTests
+ * \brief Test problem for the incompressible one-phase model:
+ *
+ * Can be run as <tt>./test_box1pfv</tt> or
+ * <tt>./test_cc1pfv</tt>
+ */
+template<class TypeTag>
+class OnePTestProblemInternalDirichlet : public OnePTestProblem<TypeTag>
+{
+ using ParentType = OnePTestProblem<TypeTag>;
+ using GridView = GetPropType<TypeTag, Properties::GridView>;
+ using Element = typename GridView::template Codim<0>::Entity;
+ using Scalar = GetPropType<TypeTag, Properties::Scalar>;
+ using PrimaryVariables = GetPropType<TypeTag, Properties::PrimaryVariables>;
+ using NeumannValues = GetPropType<TypeTag, Properties::NumEqVector>;
+ using BoundaryTypes = GetPropType<TypeTag, Properties::BoundaryTypes>;
+ using FVGridGeometry = GetPropType<TypeTag, Properties::FVGridGeometry>;
+ using SubControlVolume = typename FVGridGeometry::SubControlVolume;
+ using GlobalPosition = typename Element::Geometry::GlobalCoordinate;
+
+public:
+ OnePTestProblemInternalDirichlet(std::shared_ptr<const FVGridGeometry> fvGridGeometry)
+ : ParentType(fvGridGeometry)
+ {}
+
+ /*!
+ * \brief Specifies which kind of boundary condition should be
+ * used for which equation on a given boundary control volume.
+ *
+ * \param globalPos The position of the center of the finite volume
+ */
+ BoundaryTypes boundaryTypesAtPos(const GlobalPosition &globalPos) const
+ {
+ BoundaryTypes values;
+ values.setAllNeumann();
+ return values;
+ }
+
+ /*!
+ * \brief Evaluate the boundary conditions for a neumann
+ * boundary segment.
+ *
+ * \param globalPos The position of the boundary face's integration point in global coordinates
+ *
+ * Negative values mean influx.
+ * E.g. for the mass balance that would be the mass flux in \f$ [ kg / (m^2 \cdot s)] \f$.
+ */
+ NeumannValues neumannAtPos(const GlobalPosition& globalPos) const
+ {
+ const auto& gg = this->fvGridGeometry();
+ if (globalPos[0] < gg.bBoxMin()[0] + eps_)
+ return NeumannValues(1e3);
+ else if (globalPos[1] < gg.bBoxMin()[1] + eps_)
+ return NeumannValues(-1e3);
+ else
+ return NeumannValues(0.0);
+ }
+
+ //! Enable internal Dirichlet constraints
+ static constexpr bool enableInternalDirichletConstraints()
+ { return true; }
+
+ bool hasInternalDirichletConstraint(const Element& element, const SubControlVolume& scv) const
+ {
+ // the pure Neumann problem is only defined up to a constant
+ // we create a well-posed problem by fixing the pressure at one dof in the middle of the domain
+ return (scv.dofIndex() == static_cast<std::size_t>(this->fvGridGeometry().numDofs()/2));
+ }
+
+ PrimaryVariables internalDirichlet(const Element& element, const SubControlVolume& scv) const
+ { return PrimaryVariables(1e5); }
+
+private:
+ static constexpr Scalar eps_ = 1.5e-7;
+};
+
+} // end namespace Dumux
+
+#endif
diff --git a/test/references/test_1p_internaldirichlet_box-reference.vtu b/test/references/test_1p_internaldirichlet_box-reference.vtu
new file mode 100644
index 0000000000000000000000000000000000000000..c981becbf05c2df21a05ce76ac9f31efe784de37
--- /dev/null
+++ b/test/references/test_1p_internaldirichlet_box-reference.vtu
@@ -0,0 +1,130 @@
+<?xml version="1.0"?>
+<VTKFile type="UnstructuredGrid" version="0.1" byte_order="LittleEndian">
+ <UnstructuredGrid>
+ <Piece NumberOfCells="100" NumberOfPoints="121">
+ <PointData Scalars="p">
+ <DataArray type="Float32" Name="p" NumberOfComponents="1" format="ascii">
+ 100000 1.24072e+06 -1.04072e+06 100000 2.6305e+06 1.55452e+06 4.13529e+06 3.19796e+06 5.53808e+06 4.69269e+06 6.66643e+06 5.84341e+06
+ 7.45365e+06 6.62732e+06 7.91318e+06 7.04425e+06 8.12384e+06 7.18575e+06 8.19706e+06 7.21704e+06 8.2132e+06 7.22549e+06 -2.4305e+06 -1.35452e+06
+ 100000 2.57578e+06 4.19043e+06 5.39504e+06 6.17581e+06 6.51896e+06 6.28733e+06 6.25484e+06 6.25379e+06 -3.93529e+06 -2.99796e+06 -2.37578e+06
+ 100000 2.57651e+06 4.04084e+06 4.98735e+06 5.43331e+06 5.27427e+06 5.28578e+06 5.28474e+06 -5.33808e+06 -4.49269e+06 -3.99043e+06 -2.37651e+06
+ 100000 1.8383e+06 3.07381e+06 3.95787e+06 4.31876e+06 4.3135e+06 4.31476e+06 -6.46643e+06 -5.64341e+06 -5.19504e+06 -3.84084e+06 -1.6383e+06
+ 100000 1.50469e+06 2.73511e+06 3.33092e+06 3.34248e+06 3.34855e+06 -7.25365e+06 -6.42732e+06 -5.97581e+06 -4.78735e+06 -2.87381e+06 -1.30469e+06
+ 100000 1.54703e+06 2.33338e+06 2.37887e+06 2.39919e+06 -7.71318e+06 -6.84425e+06 -6.31896e+06 -5.23331e+06 -3.75787e+06 -2.53511e+06 -1.34703e+06
+ 100000 1.33109e+06 1.43949e+06 1.50685e+06 -7.92384e+06 -6.98575e+06 -6.08733e+06 -5.07427e+06 -4.11876e+06 -3.13092e+06 -2.13338e+06 -1.13109e+06
+ 100000 623614 760454 -7.99706e+06 -7.01704e+06 -6.05484e+06 -5.08578e+06 -4.1135e+06 -3.14248e+06 -2.17887e+06 -1.23949e+06 -423614
+ 100000 269152 -8.0132e+06 -7.02549e+06 -6.05379e+06 -5.08474e+06 -4.11476e+06 -3.14855e+06 -2.19919e+06 -1.30685e+06 -560454 -69152.5
+ 100000
+ </DataArray>
+ </PointData>
+ <CellData Scalars="process rank">
+ <DataArray type="Float32" Name="process rank" NumberOfComponents="1" format="ascii">
+ 0 0 0 0 0 0 0 0 0 0 0 0
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+ 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0 0 0 0 0 0 0 0 0
+ 0 0 0 0
+ </DataArray>
+ </CellData>
+ <Points>
+ <DataArray type="Float32" Name="Coordinates" NumberOfComponents="3" format="ascii">
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+ </Points>
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+ <DataArray type="Int32" Name="offsets" NumberOfComponents="1" format="ascii">
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+ </DataArray>
+ <DataArray type="UInt8" Name="types" NumberOfComponents="1" format="ascii">
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+ 9 9 9 9
+ </DataArray>
+ </Cells>
+ </Piece>
+ </UnstructuredGrid>
+</VTKFile>
diff --git a/test/references/test_1p_internaldirichlet_tpfa-reference.vtu b/test/references/test_1p_internaldirichlet_tpfa-reference.vtu
new file mode 100644
index 0000000000000000000000000000000000000000..8fcb11f9e72471ee32dd9b4750377e2115938037
--- /dev/null
+++ b/test/references/test_1p_internaldirichlet_tpfa-reference.vtu
@@ -0,0 +1,126 @@
+<?xml version="1.0"?>
+<VTKFile type="UnstructuredGrid" version="0.1" byte_order="LittleEndian">
+ <UnstructuredGrid>
+ <Piece NumberOfCells="100" NumberOfPoints="121">
+ <CellData Scalars="p">
+ <DataArray type="Float32" Name="p" NumberOfComponents="1" format="ascii">
+ 8.95261e+06 1.05793e+07 1.28325e+07 1.49081e+07 1.65855e+07 1.78052e+07 1.85563e+07 1.88584e+07 1.88036e+07 1.87437e+07 7.32597e+06 8.95261e+06
+ 1.20103e+07 1.43062e+07 1.60432e+07 1.72739e+07 1.80054e+07 1.82151e+07 1.78089e+07 1.76838e+07 5.07269e+06 5.89496e+06 8.95261e+06 1.21396e+07
+ 1.42231e+07 1.56494e+07 1.65325e+07 1.68372e+07 1.6533e+07 1.64987e+07 2.99715e+06 3.59902e+06 5.76567e+06 8.95261e+06 1.1276e+07 1.29759e+07
+ 1.41945e+07 1.50158e+07 1.52853e+07 1.52794e+07 1.31973e+06 1.86205e+06 3.68216e+06 6.62918e+06 8.95261e+06 1.07837e+07 1.22535e+07 1.34824e+07
+ 1.40488e+07 1.40542e+07 100000 631357 2.25581e+06 4.92933e+06 7.12156e+06 8.95261e+06 1.05537e+07 1.2056e+07 1.28181e+07 1.28345e+07
+ -651091 -100147 1.3727e+06 3.71077e+06 5.65168e+06 7.35153e+06 8.95261e+06 1.06229e+07 1.15861e+07 1.16312e+07 -953124 -309874
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+ 6.27406e+06 7.43234e+06 8.44586e+06 8.95261e+06
+ </DataArray>
+ <DataArray type="Float32" Name="process rank" NumberOfComponents="1" format="ascii">
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