[coupling-ff-pm] Add interface exercise

exercises/CMakeLists.txt

@@ -2,6 +2,7 @@
 add_custom_target(test_exercises)
 
 add_subdirectory(exercise-basic)
+add_subdirectory(exercise-coupling-ff-pm)
 add_subdirectory(exercise-runtimeparams)
 add_subdirectory(exercise-grids)
 add_subdirectory(exercise-properties)

exercises/exercise-coupling-ff-pm/CMakeLists.txt

+add_subdirectory(interface)

exercises/exercise-coupling-ff-pm/README.md

+# Exercise #5 (DuMuX course)
+
+The aim of this exercise is to get familiar with the set up of coupled free flow/porous medium flow problems.
+
+## Problem set-up
+
+The model domain consists of two non-overlapping subdomains.
+Free flow is modeled in the upper subdomain, while the lower subdomain models a flow in a porous medium.
+Both single-phase flow and two-phase flow will be considered in the porous domain.
+
+![](../extradoc/exercise5_setup.png)
+
+
+
+
+### 0. Getting familiar with the code
+
+* Navigate to the directory `exercises/exercise-coupling-ff-pm`
+
+There are three sub folders: `interface`, `models` and `turbulence`.
+
+The problem-related files for this exercise are:
+* Three __main files__ for the three sub-tasks :`ex_interface_coupling_ff-pm.cc`, `ex2_coupling_ff-pm.cc`, `ex2_coupling_ff-pm.cc`,
+* Three __free flow problem files__: `ex_interface_ffproblem.hh`, `ex2_ffproblem.hh`, `ex3_ffproblem.hh`
+* Three __porous medium flow problem files__: `ex_interface_pmproblem.hh`, `ex2_pmproblem.hh`, `ex3_pmproblem.hh`
+* The __input files__: ...
+* The __spatial parameters files__: `1pspatialparams.hh`, `2pspatialparams.hh`
+
+
+In the main file, `TypeTags` for both submodels are defined. The same applies for types such as `GridManager`, `FVGridGeometry`, `Problem`, etc.. Since we use a monolithic coupling scheme, there is only one `Assembler` and one
+`NewtonSolver`.
+
+The problem files very much look like "regular", uncoupled ones with the exception that they hold a pointer to the `CouplingManager` which allows to evaluate the coupling conditions and to exchange information between the coupled models. The coupling conditions are realized technically in terms of boundary condition. For instance, in lines 178 and 179
+in `ex_interface_ffproblem.hh`, `couplingNeumann` boundary conditions are set, which means that the free flow models evaluates the
+mass and momentum fluxes coming from the porous domain and uses these values as boundary conditions at the interface.
+
+Note the certain checks are performed when combining different models, e.g., the fluid system has to be the same for both domains.
+
+
+We will use a staggered grid for the free flow and a cell-centered finite volume method for the porous medium.
+Keep in mind that the staggered grid implementation distinguishes between face variables (velocity components) and
+cell center variables (all other variables), therefore in some cases either the `stokesCellCenterIdx`
+or the `stokesFaceIdx` is used respectively, while for the porous medium all variables can be accessed with `darcyIdx`.
+
+The main part of the Stokes/Darcy coupling is implemented in the folder `dumux/multidomain/boundary/stokesdarcy`.
+
+__Task__:
+Take a closer look at the Stokes/Darcy coupling files and try to answer the following questions before moving to the next part of the exercise:
+* Where are the elements at the coupling interface paired with their respective coupling partner elements of the other domain?
+* Where are the coupling conditions implemented and evaluated?
+* You will find four implementations of the `massCouplingCondition`. What are the differences between them?
+* Which data is stored and updated regularly such that it is accessible from the respective other domain?
+* How can we access volume variables like density and pressure from the respective other domain?
+* ...
+
+### 1. Changing the interface
+
+In this part of the exercise, a simple coupled system consisting of a one-phase (1p) free flow and a one-phase flow in a porous medium is set up. Both subproblems have no-flow boundaries at the sides.
+A velocity profile is set on the upper free flow boundary, which leads to a vertical flow into the porous medium.
+
+* We will first change the flow direction such that the free flow is parallel to the porous medium.
+* Afterwards, the Beavers-Joseph-Saffman condition will be used as an interface condition for the tangential momentum transfer.
+* Last, we change the flat interface between the two domains to a wave-shaped one.
+
+__Tasks__:
+
+Open the file `ex_interface_ffproblem.hh` and navigate to line 155, where the types of boundary condition are set.
+Instead of applying a fixed velocity profile at the top of the domain, we want to use fixed pressure boundary conditions
+at the left and right side of the free flow domain, while the top represents an impermeable wall.
+
+__Change the flow direction__
+
+Set a Dirichlet boundary condition for the pressure at the left and ride side of the domain:
+``` cpp
+if(onLeftBoundary_(globalPos) || onRightBoundary_(globalPos))
+    values.setDirichlet(Indices::pressureIdx);
+
+```
+
+Set a Dirichlet boundary condition for the velocities at the top:
+``` cpp
+if(onUpperBoundary_(globalPos))
+{
+    values.setDirichlet(Indices::velocityXIdx);
+    values.setDirichlet(Indices::velocityYIdx);
+}
+```
+
+Keep the coupling boundary condition:
+``` cpp
+if(couplingManager().isCoupledEntity(CouplingManager::stokesIdx, scvf))
+{
+    values.setCouplingNeumann(Indices::conti0EqIdx);
+    values.setCouplingNeumann(Indices::momentumYBalanceIdx);
+}
+```
+
+Having changed the types of boundary conditions, we must now assign the correct values for them.
+
+Set a no-slip, no-flow condition for the velocity at the top:
+``` cpp
+values[Indices::velocityXIdx] = 0.0;
+values[Indices::velocityYIdx] = 0.0;
+```
+Apply a fixed pressure difference between the inlet and outlet, e.g.:
+``` cpp
+if(onLeftBoundary_(globalPos))
+    values[Indices::pressureIdx] = deltaP_;
+if(onRightBoundary_(globalPos))
+    values[Indices::pressureIdx] = 0.0;
+```
+
+For changing the flow direction, the boundary conditions for the porous medium have to be changed as well.
+
+Use Neumann no-flow boundaries everywhere, keep the coupling conditions.
+``` cpp
+values.setAllNeumann();
+
+if (couplingManager().isCoupledEntity(CouplingManager::darcyIdx, scvf))
+    values.setAllCouplingNeumann();
+```
+
+__Include slip-condition__
+
+This should make the flow go from left to right. However, we are still missing one important feature:
+the Beavers-Joseph-Saffman slip condition at the interface:
+
+
+$`\frac{\partial v_x}{\partial y} = \frac{\alpha}{\sqrt K} (v_x - q_{pm})\quad`$ at $`\quad y=0`$
+
+
+with  $`\quad q_{pm}=0`$.
+
+The include this, we just set:
+``` cpp
+values.setBJS(Indices::momentumXBalanceIdx);
+```
+
+at the position where the coupling boundary conditions are set in `ex_interface_ffproblem.hh`.
+
+To check if the simulation behaves as expected, we can compare the velocity profile $`v_x(y)`$ with the analytical solution provided by [Beavers and Joseph (1967)](https://doi.org/10.1017/S0022112067001375).
+For doing so, we uncomment line 212
+```cpp
+stokesVtkWriter.addField(stokesProblem->getAnalyticalVelocityX(), "analyticalV_x");
+```
+
+in `ex_interface_coupling_ff-pm.cc`. After re-compiling and re-running the executable, we should be able to see also
+the analytical solution of $`v_x`$ on the free flow domain. Play around with the grid resolution to see how that affects the velocity profile.
+
+__Cange shape of interface__
+
+Now we want to include a non-flat interface between the two domains. We use `dune-subgrid` to construct
+two grids for the two domains from one common host grid. Comment out lines 93-103 in `ex_interface_coupling_ff-pm.cc` and comment lines 112-147 in the same file. This will instantiate a host grid and define two helper lambda functions that are used to choose elements from to host grid for the respective sub grid. In the given case,
+the domain is split in two haves, separated by a sinusoidal interface.
+
+```cpp
+auto elementSelectorStokes = [&](const auto& element)
+{
+    double interface = params.amplitude * std::sin(( element.geometry().center()[0] -params.offset) / params.scaling * 2.0 * M_PI) + params.baseline;
+    return element.geometry().center()[1] > interface;
+};
+
+auto elementSelectorDarcy = [&](const auto& element)
+{
+    double interface  =  params.amplitude * std::sin(( element.geometry().center()[0] - params.offset) / params.scaling * 2.0 * M_PI) + params.baseline;
+    return element.geometry().center()[1] < interface;
+};
+```
+
+Make sure to comment in line 30 in both problem files
+```cpp
+#include <dumux/io/grid/subgridgridcreator.hh>
+```
+
+and do the changes in the respective lines for the `Grid` property.
+
+The problem should no compile and run. However, an error occurs due to the coupling conditions.
+So far, we assumed a flat interface, therefore the normal momentum coupling condition
+
+ $`[\sigma \cdot \mathbf{n}]^{FF} = p^{PM}`$
+
+ was always set for a fixed $`\mathbf{n} = (0,1)^T`$. We need to account for the curvature of the interface and thus replace
+ ```cpp
+values.setCouplingNeumann(Indices::momentumYBalanceIdx);
+ ```
+ with
+ ```cpp
+values.setCouplingNeumann(scvf.directionIndex());
+ ```
+The same if true for the BJS condition, however, here we need to consider the tangential direction:
+```cpp
+values.setCouplingNeumann(1 - scvf.directionIndex());
+```
+
+The final result should look something like this:
+![](../extradoc/ex_ff-pm-wave-interface.png)
+
+
+### 3. Change the models
+
+1p2c/1p2c -->1p2c/2p2c
+
+__Task__:
+
+...
+
+### 4. ...

exercises/exercise-coupling-ff-pm/interface/1pspatialparams.hh

+// -*- 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 2 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 spatial parameters class for the test problem using the 1p cc model
+ */
+#ifndef DUMUX_1P_TEST_SPATIALPARAMS_HH
+#define DUMUX_1P_TEST_SPATIALPARAMS_HH
+
+#include <dumux/material/spatialparams/fv1p.hh>
+
+namespace Dumux
+{
+
+/*!
+ * \ingroup OnePModel
+ * \ingroup ImplicitTestProblems
+ *
+ * \brief The spatial parameters class for the test problem using the
+ *        1p cc model
+ */
+template<class TypeTag>
+class OnePSpatialParams
+: public FVSpatialParamsOneP<typename GET_PROP_TYPE(TypeTag, FVGridGeometry),
+                             typename GET_PROP_TYPE(TypeTag, Scalar),
+                             OnePSpatialParams<TypeTag>>
+{
+    using FVGridGeometry = typename GET_PROP_TYPE(TypeTag, FVGridGeometry);
+    using Scalar = typename GET_PROP_TYPE(TypeTag, Scalar);
+    using Problem = typename GET_PROP_TYPE(TypeTag, Problem);
+    using GridView = typename GET_PROP_TYPE(TypeTag, GridView);
+    using ParentType = FVSpatialParamsOneP<FVGridGeometry, Scalar, OnePSpatialParams<TypeTag>>;
+
+    using Element = typename GridView::template Codim<0>::Entity;
+    using GlobalPosition = typename Element::Geometry::GlobalCoordinate;
+
+public:
+    // export permeability type
+    using PermeabilityType = Scalar;
+
+    OnePSpatialParams(std::shared_ptr<const FVGridGeometry> fvGridGeometry)
+        : ParentType(fvGridGeometry)
+    {
+        permeability_ = getParam<Scalar>("Darcy.SpatialParams.Permeability");
+        alphaBJ_ = getParam<Scalar>("Darcy.SpatialParams.AlphaBeaversJoseph");
+    }
+
+    /*!
+     * \brief Function for defining the (intrinsic) permeability \f$[m^2]\f$.
+     *
+     * \param globalPos The global position
+     * \return the intrinsic permeability
+     */
+    PermeabilityType permeabilityAtPos(const GlobalPosition& globalPos) const
+    { return permeability_; }
+
+    /*! \brief Define the porosity in [-].
+     *
+     * \param globalPos The global position
+     */
+    Scalar porosityAtPos(const GlobalPosition& globalPos) const
+    { return 0.4; }
+
+    /*! \brief Define the Beavers-Joseph coefficient in [-].
+     *
+     * \param globalPos The global position
+     */
+    Scalar beaversJosephCoeffAtPos(const GlobalPosition& globalPos) const
+    { return alphaBJ_; }
+
+
+private:
+    Scalar permeability_;
+    Scalar alphaBJ_;
+};
+
+} // end namespace
+
+#endif

exercises/exercise-coupling-ff-pm/interface/CMakeLists.txt

+add_input_file_links()
+
+# executables for ex_interface_coupling_ff-pm
+dune_add_test(NAME ex_interface_coupling_ff-pm
+              SOURCES ex_interface_coupling_ff-pm.cc
+              CMD_ARGS ex_interface_coupling_ff-pm.input)
+
+# add tutorial to the common target
+add_dependencies(test_exercises ex_interface_coupling_ff-pm)

exercises/exercise-coupling-ff-pm/interface/ex_interface_coupling_ff-pm.cc

+// -*- 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 2 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
+ *
+ * \brief A test problem for the coupled Stokes/Darcy problem (1p)
+ */
+#include <config.h>
+
+#include <iostream>
+
+#include <dune/common/parallel/mpihelper.hh>
+#include <dune/istl/io.hh>
+
+#include <dumux/common/properties.hh>
+#include <dumux/common/parameters.hh>
+#include <dumux/common/dumuxmessage.hh>
+#include <dumux/linear/seqsolverbackend.hh>
+#include <dumux/assembly/fvassembler.hh>
+#include <dumux/assembly/diffmethod.hh>
+#include <dumux/discretization/methods.hh>
+#include <dumux/io/vtkoutputmodule.hh>
+#include <dumux/io/staggeredvtkoutputmodule.hh>
+#include <dumux/io/grid/gridmanager.hh>
+
+#include <dumux/multidomain/staggeredtraits.hh>
+#include <dumux/multidomain/fvassembler.hh>
+#include <dumux/multidomain/newtonsolver.hh>
+
+#include <dumux/multidomain/boundary/stokesdarcy/couplingmanager.hh>
+
+#include "ex_interface_pmproblem.hh"
+#include "ex_interface_ffproblem.hh"
+
+namespace Dumux {
+namespace Properties {
+
+SET_PROP(StokesOnePTypeTag, CouplingManager)
+{
+    using Traits = StaggeredMultiDomainTraits<TypeTag, TypeTag, TTAG(DarcyOnePTypeTag)>;
+    using type = Dumux::StokesDarcyCouplingManager<Traits>;
+};
+
+SET_PROP(DarcyOnePTypeTag, CouplingManager)
+{
+    using Traits = StaggeredMultiDomainTraits<TTAG(StokesOnePTypeTag), TTAG(StokesOnePTypeTag), TypeTag>;
+    using type = Dumux::StokesDarcyCouplingManager<Traits>;
+};
+
+} // end namespace Properties
+} // end namespace Dumux
+
+int main(int argc, char** argv) try
+{
+    using namespace Dumux;
+
+    // initialize MPI, finalize is done automatically on exit
+    const auto& mpiHelper = Dune::MPIHelper::instance(argc, argv);
+
+    // print dumux start message
+    if (mpiHelper.rank() == 0)
+        DumuxMessage::print(/*firstCall=*/true);
+
+    // parse command line arguments and input file
+    Parameters::init(argc, argv);
+
+    // Define the sub problem type tags
+    using StokesTypeTag = TTAG(StokesOnePTypeTag);
+    using DarcyTypeTag = TTAG(DarcyOnePTypeTag);
+
+
+
+    // ******************** comment-out this section for the last exercise **************** //
+
+    // create two individual grids (from the given grid file or the input file)
+    // for both sub-domains
+    using DarcyGridManager = Dumux::GridManager<typename GET_PROP_TYPE(DarcyTypeTag, Grid)>;
+    DarcyGridManager darcyGridManager;
+    darcyGridManager.init("Darcy"); // pass parameter group
+
+    using StokesGridManager = Dumux::GridManager<typename GET_PROP_TYPE(StokesTypeTag, Grid)>;
+    StokesGridManager stokesGridManager;
+    stokesGridManager.init("Stokes"); // pass parameter group
+
+    // we compute on the leaf grid view
+    const auto& darcyGridView = darcyGridManager.grid().leafGridView();
+    const auto& stokesGridView = stokesGridManager.grid().leafGridView();
+
+    // ************************************************************************************ //
+
+
+    // ******************** uncomment this section for the last exercise ****************** //
+
+    // // use dune-subgrid to create the individual grids
+    // static constexpr int dim = 2;
+    // using HostGrid = Dune::YaspGrid<2, Dune::TensorProductCoordinates<double, dim> >;
+    // using HostGridManager = Dumux::GridManager<HostGrid>;
+    // HostGridManager hostGridManager;
+    // hostGridManager.init();
+    // auto& hostGrid = hostGridManager.grid();
+    //
+    // struct Params
+    // {
+    //     double amplitude = getParam<double>("Grid.Amplitude");
+    //     double baseline = getParam<double>("Grid.Baseline");
+    //     double offset = getParam<double>("Grid.Offset");
+    //     double scaling = getParam<double>("Grid.Scaling");
+    // };
+    //
+    // Params params;
+    //
+    // auto elementSelectorStokes = [&](const auto& element)
+    // {
+    //     double interface = params.amplitude * std::sin(( element.geometry().center()[0] -params.offset) / params.scaling * 2.0 * M_PI) + params.baseline;
+    //     return element.geometry().center()[1] > interface;
+    // };
+    //
+    // auto elementSelectorDarcy = [&](const auto& element)
+    // {
+    //     double interface  =  params.amplitude * std::sin(( element.geometry().center()[0] - params.offset) / params.scaling * 2.0 * M_PI) + params.baseline;
+    //     return element.geometry().center()[1] < interface;
+    // };
+    //
+    // // subgrid Pointer
+    // auto stokesGridPtr = SubgridGridCreator<HostGrid>::makeGrid(hostGrid, elementSelectorStokes, "Stokes");
+    // auto darcyGridPtr = SubgridGridCreator<HostGrid>::makeGrid(hostGrid, elementSelectorDarcy, "Darcy");
+    //
+    // // we compute on the leaf grid view
+    // const auto& darcyGridView = darcyGridPtr->leafGridView();
+    // const auto& stokesGridView = stokesGridPtr->leafGridView();
+
+    // ************************************************************************************ //
+
+
+    // create the finite volume grid geometry
+    using StokesFVGridGeometry = typename GET_PROP_TYPE(StokesTypeTag, FVGridGeometry);
+    auto stokesFvGridGeometry = std::make_shared<StokesFVGridGeometry>(stokesGridView);
+    stokesFvGridGeometry->update();
+    using DarcyFVGridGeometry = typename GET_PROP_TYPE(DarcyTypeTag, FVGridGeometry);
+    auto darcyFvGridGeometry = std::make_shared<DarcyFVGridGeometry>(darcyGridView);
+    darcyFvGridGeometry->update();
+
+    using Traits = StaggeredMultiDomainTraits<StokesTypeTag, StokesTypeTag, DarcyTypeTag>;
+
+    // the coupling manager
+    using CouplingManager = StokesDarcyCouplingManager<Traits>;
+    auto couplingManager = std::make_shared<CouplingManager>(stokesFvGridGeometry, darcyFvGridGeometry);
+
+    // the indices
+    constexpr auto stokesCellCenterIdx = CouplingManager::stokesCellCenterIdx;
+    constexpr auto stokesFaceIdx = CouplingManager::stokesFaceIdx;
+    constexpr auto darcyIdx = CouplingManager::darcyIdx;
+
+    // the problem (initial and boundary conditions)
+    using StokesProblem = typename GET_PROP_TYPE(StokesTypeTag, Problem);
+    auto stokesProblem = std::make_shared<StokesProblem>(stokesFvGridGeometry, couplingManager);
+    using DarcyProblem = typename GET_PROP_TYPE(DarcyTypeTag, Problem);
+    auto darcyProblem = std::make_shared<DarcyProblem>(darcyFvGridGeometry, couplingManager);
+
+    // the solution vector
+    Traits::SolutionVector sol;
+    sol[stokesCellCenterIdx].resize(stokesFvGridGeometry->numCellCenterDofs());
+    sol[stokesFaceIdx].resize(stokesFvGridGeometry->numFaceDofs());
+    sol[darcyIdx].resize(darcyFvGridGeometry->numDofs());
+
+    const auto& cellCenterSol = sol[stokesCellCenterIdx];
+    const auto& faceSol = sol[stokesFaceIdx];
+
+    // apply initial solution for instationary problems
+    typename GET_PROP_TYPE(StokesTypeTag, SolutionVector) stokesSol;
+    std::get<0>(stokesSol) = cellCenterSol;
+    std::get<1>(stokesSol) = faceSol;
+    stokesProblem->applyInitialSolution(stokesSol);
+    sol[stokesCellCenterIdx] = stokesSol[stokesCellCenterIdx];
+    sol[stokesFaceIdx] = stokesSol[stokesFaceIdx];
+
+    couplingManager->init(stokesProblem, darcyProblem, sol);
+
+    // the grid variables
+    using StokesGridVariables = typename GET_PROP_TYPE(StokesTypeTag, GridVariables);
+    auto stokesGridVariables = std::make_shared<StokesGridVariables>(stokesProblem, stokesFvGridGeometry);
+    stokesGridVariables->init(stokesSol);
+    using DarcyGridVariables = typename GET_PROP_TYPE(DarcyTypeTag, GridVariables);
+    auto darcyGridVariables = std::make_shared<DarcyGridVariables>(darcyProblem, darcyFvGridGeometry);
+    darcyGridVariables->init(sol[darcyIdx]);
+
+    // intialize the vtk output module
+    const auto stokesName = getParam<std::string>("Problem.Name") + "_" + stokesProblem->name();
+    const auto darcyName = getParam<std::string>("Problem.Name") + "_" + darcyProblem->name();
+
+    StaggeredVtkOutputModule<StokesTypeTag> stokesVtkWriter(*stokesProblem, *stokesFvGridGeometry, *stokesGridVariables, stokesSol, stokesName);
+    GET_PROP_TYPE(StokesTypeTag, VtkOutputFields)::init(stokesVtkWriter);
+
+    //****** uncomment the add analytical solution of v_x *****//
+    // stokesVtkWriter.addField(stokesProblem->getAnalyticalVelocityX(), "analyticalV_x");
+
+    stokesVtkWriter.write(0.0);
+
+    VtkOutputModule<DarcyTypeTag> darcyVtkWriter(*darcyProblem, *darcyFvGridGeometry, *darcyGridVariables, sol[darcyIdx], darcyName);
+    GET_PROP_TYPE(DarcyTypeTag, VtkOutputFields)::init(darcyVtkWriter);
+    darcyVtkWriter.write(0.0);
+
+    // the assembler for a stationary problem
+    using Assembler = MultiDomainFVAssembler<Traits, CouplingManager, DiffMethod::numeric>;
+    auto assembler = std::make_shared<Assembler>(std::make_tuple(stokesProblem, stokesProblem, darcyProblem),
+                                                 std::make_tuple(stokesFvGridGeometry->cellCenterFVGridGeometryPtr(),
+                                                                 stokesFvGridGeometry->faceFVGridGeometryPtr(),
+                                                                 darcyFvGridGeometry),
+                                                 std::make_tuple(stokesGridVariables->cellCenterGridVariablesPtr(),
+                                                                 stokesGridVariables->faceGridVariablesPtr(),
+                                                                 darcyGridVariables),
+                                                 couplingManager);
+
+    // the linear solver
+    using LinearSolver = UMFPackBackend;
+    auto linearSolver = std::make_shared<LinearSolver>();
+
+    // the non-linear solver
+    using NewtonSolver = MultiDomainNewtonSolver<Assembler, LinearSolver, CouplingManager>;
+    NewtonSolver nonLinearSolver(assembler, linearSolver, couplingManager);
+
+    // solve the non-linear system
+    nonLinearSolver.solve(sol);
+
+    // write vtk output
+    stokesVtkWriter.write(1.0);
+    darcyVtkWriter.write(1.0);
+
+    ////////////////////////////////////////////////////////////
+    // finalize, print dumux message to say goodbye
+    ////////////////////////////////////////////////////////////
+
+    // print dumux end message
+    if (mpiHelper.rank() == 0)
+    {
+        Parameters::print();
+        DumuxMessage::print(/*firstCall=*/false);
+    }
+
+    return 0;
+} // end main
+catch (Dumux::ParameterException &e)
+{
+    std::cerr << std::endl << e << " ---> Abort!" << std::endl;
+    return 1;
+}
+catch (Dune::DGFException & e)