subdomainboxlocalassembler.hh

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#ifndef DUMUX_MULTIDOMAIN_BOX_LOCAL_ASSEMBLER_HH
#define DUMUX_MULTIDOMAIN_BOX_LOCAL_ASSEMBLER_HH
 
#include <dune/common/reservedvector.hh>
#include <dune/common/indices.hh>
#include <dune/common/hybridutilities.hh>
#include <dune/grid/common/gridenums.hh> // for GhostEntity
#include <dune/istl/matrixindexset.hh>
 
#include <dumux/common/reservedblockvector.hh>
#include <dumux/common/properties.hh>
#include <dumux/common/parameters.hh>
#include <dumux/common/numericdifferentiation.hh>
#include <dumux/assembly/numericepsilon.hh>
#include <dumux/assembly/diffmethod.hh>
#include <dumux/assembly/fvlocalassemblerbase.hh>
 
namespace Dumux {
 
template<std::size_t id, class TypeTag, class Assembler, class Implementation, bool implicit>
class SubDomainBoxLocalAssemblerBase : public FVLocalAssemblerBase<TypeTag, Assembler, Implementation, implicit>
{
    using ParentType = FVLocalAssemblerBase<TypeTag, Assembler,Implementation, implicit>;
 
    using Problem = GetPropType<TypeTag, Properties::Problem>;
    using LocalResidualValues = GetPropType<TypeTag, Properties::NumEqVector>;
    using ElementResidualVector = typename ParentType::LocalResidual::ElementResidualVector;
    using JacobianMatrix = GetPropType<TypeTag, Properties::JacobianMatrix>;
    using SolutionVector = typename Assembler::SolutionVector;
    using SubSolutionVector = GetPropType<TypeTag, Properties::SolutionVector>;
    using ElementBoundaryTypes = GetPropType<TypeTag, Properties::ElementBoundaryTypes>;
 
    using GridVariables = GetPropType<TypeTag, Properties::GridVariables>;
    using GridVolumeVariables = typename GridVariables::GridVolumeVariables;
    using ElementVolumeVariables = typename GridVolumeVariables::LocalView;
    using ElementFluxVariablesCache = typename GridVariables::GridFluxVariablesCache::LocalView;
    using Scalar = typename GridVariables::Scalar;
 
    using GridGeometry = typename GridVariables::GridGeometry;
    using FVElementGeometry = typename GridGeometry::LocalView;
    using SubControlVolume = typename GridGeometry::SubControlVolume;
    using SubControlVolumeFace = typename GridGeometry::SubControlVolumeFace;
    using GridView = typename GridGeometry::GridView;
    using Element = typename GridView::template Codim<0>::Entity;
 
    using CouplingManager = typename Assembler::CouplingManager;
 
    static constexpr auto numEq = GetPropType<TypeTag, Properties::ModelTraits>::numEq();
 
public:
    static constexpr auto domainId = typename Dune::index_constant<id>();
    using ParentType::ParentType;
 
    // the constructor
    explicit SubDomainBoxLocalAssemblerBase(const Assembler& assembler,
                                            const Element& element,
                                            const SolutionVector& curSol,
                                            CouplingManager& couplingManager)
    : ParentType(assembler,
                 element,
                 curSol,
                 localView(assembler.gridGeometry(domainId)),
                 localView(assembler.gridVariables(domainId).curGridVolVars()),
                 localView(assembler.gridVariables(domainId).prevGridVolVars()),
                 localView(assembler.gridVariables(domainId).gridFluxVarsCache()),
                 assembler.localResidual(domainId),
                 (element.partitionType() == Dune::GhostEntity))
    , couplingManager_(couplingManager)
    {}
 
    template<class JacobianMatrixRow, class GridVariablesTuple>
    void assembleJacobianAndResidual(JacobianMatrixRow& jacRow, SubSolutionVector& res, GridVariablesTuple& gridVariables)
    {
        this->asImp_().bindLocalViews();
        this->elemBcTypes().update(problem(), this->element(), this->fvGeometry());
 
        // for the diagonal jacobian block
        // forward to the internal implementation
        const auto residual = this->asImp_().assembleJacobianAndResidualImpl(jacRow[domainId], *std::get<domainId>(gridVariables));
 
        // update the residual vector
        for (const auto& scv : scvs(this->fvGeometry()))
            res[scv.dofIndex()] += residual[scv.localDofIndex()];
 
        // assemble the coupling blocks
        using namespace Dune::Hybrid;
        forEach(integralRange(Dune::Hybrid::size(jacRow)), [&](auto&& i)
        {
            if (i != id)
                this->assembleJacobianCoupling(i, jacRow, residual, gridVariables);
        });
 
        // lambda for the incorporation of Dirichlet Bcs
        auto applyDirichlet = [&] (const auto& scvI,
                                   const auto& dirichletValues,
                                   const auto eqIdx,
                                   const auto pvIdx)
        {
            res[scvI.dofIndex()][eqIdx] = this->curElemVolVars()[scvI].priVars()[pvIdx] - dirichletValues[pvIdx];
 
            // in explicit schemes we only have entries on the diagonal
            // and thus don't have to do anything with off-diagonal entries
            if (implicit)
            {
                for (const auto& scvJ : scvs(this->fvGeometry()))
                    jacRow[domainId][scvI.dofIndex()][scvJ.dofIndex()][eqIdx] = 0.0;
            }
 
            jacRow[domainId][scvI.dofIndex()][scvI.dofIndex()][eqIdx][pvIdx] = 1.0;
        };
 
        // incorporate Dirichlet BCs
        this->asImp_().enforceDirichletConstraints(applyDirichlet);
    }
 
    template<std::size_t otherId, class JacRow, class GridVariables,
             typename std::enable_if_t<(otherId == id), int> = 0>
    void assembleJacobianCoupling(Dune::index_constant<otherId> domainJ, JacRow& jacRow,
                                  const ElementResidualVector& res, GridVariables& gridVariables)
    {}
 
    template<std::size_t otherId, class JacRow, class GridVariables,
             typename std::enable_if_t<(otherId != id), int> = 0>
    void assembleJacobianCoupling(Dune::index_constant<otherId> domainJ, JacRow& jacRow,
                                  const ElementResidualVector& res, GridVariables& gridVariables)
    {
        this->asImp_().assembleJacobianCoupling(domainJ, jacRow[domainJ], res, *std::get<domainId>(gridVariables));
    }
 
    void assembleResidual(SubSolutionVector& res)
    {
        this->asImp_().bindLocalViews();
        this->elemBcTypes().update(problem(), this->element(), this->fvGeometry());
 
        const auto residual = this->evalLocalResidual();
        for (const auto& scv : scvs(this->fvGeometry()))
            res[scv.dofIndex()] += residual[scv.localDofIndex()];
 
        auto applyDirichlet = [&] (const auto& scvI,
                                   const auto& dirichletValues,
                                   const auto eqIdx,
                                   const auto pvIdx)
        {
            res[scvI.dofIndex()][eqIdx] = this->curElemVolVars()[scvI].priVars()[pvIdx] - dirichletValues[pvIdx];
        };
 
        this->asImp_().enforceDirichletConstraints(applyDirichlet);
    }
 
    ElementResidualVector evalLocalSourceResidual(const Element& element, const ElementVolumeVariables& elemVolVars) const
    {
        // initialize the residual vector for all scvs in this element
        ElementResidualVector residual(this->fvGeometry().numScv());
 
        // evaluate the volume terms (storage + source terms)
        // forward to the local residual specialized for the discretization methods
        for (auto&& scv : scvs(this->fvGeometry()))
        {
            const auto& curVolVars = elemVolVars[scv];
            auto source = this->localResidual().computeSource(problem(), element, this->fvGeometry(), elemVolVars, scv);
            source *= -scv.volume()*curVolVars.extrusionFactor();
            residual[scv.localDofIndex()] = std::move(source);
        }
 
        return residual;
    }
 
    ElementResidualVector evalLocalSourceResidual(const Element& neighbor) const
    { return this->evalLocalSourceResidual(neighbor, implicit ? this->curElemVolVars() : this->prevElemVolVars()); }
 
    template<typename ApplyFunction>
    void enforceDirichletConstraints(const ApplyFunction& applyDirichlet)
    {
        // enforce Dirichlet boundary conditions
        this->asImp_().evalDirichletBoundaries(applyDirichlet);
        // take care of internal Dirichlet constraints (if enabled)
        this->asImp_().enforceInternalDirichletConstraints(applyDirichlet);
    }
 
    template< typename ApplyDirichletFunctionType >
    void evalDirichletBoundaries(ApplyDirichletFunctionType applyDirichlet)
    {
        // enforce Dirichlet boundaries by overwriting partial derivatives with 1 or 0
        // and set the residual to (privar - dirichletvalue)
        if (this->elemBcTypes().hasDirichlet())
        {
            for (const auto& scvI : scvs(this->fvGeometry()))
            {
                const auto bcTypes = this->elemBcTypes()[scvI.localDofIndex()];
                if (bcTypes.hasDirichlet())
                {
                    const auto dirichletValues = this->problem().dirichlet(this->element(), scvI);
 
                    // set the dirichlet conditions in residual and jacobian
                    for (int eqIdx = 0; eqIdx < numEq; ++eqIdx)
                    {
                        if (bcTypes.isDirichlet(eqIdx))
                        {
                            const auto pvIdx = bcTypes.eqToDirichletIndex(eqIdx);
                            assert(0 <= pvIdx && pvIdx < numEq);
                            applyDirichlet(scvI, dirichletValues, eqIdx, pvIdx);
                        }
                    }
                }
            }
        }
    }
 
    void bindLocalViews()
    {
        // get some references for convenience
        const auto& element = this->element();
        const auto& curSol = this->curSol()[domainId];
        auto&& fvGeometry = this->fvGeometry();
        auto&& curElemVolVars = this->curElemVolVars();
        auto&& elemFluxVarsCache = this->elemFluxVarsCache();
 
        // bind the caches
        couplingManager_.bindCouplingContext(domainId, element, this->assembler());
        fvGeometry.bind(element);
 
        if (implicit)
        {
            curElemVolVars.bind(element, fvGeometry, curSol);
            elemFluxVarsCache.bind(element, fvGeometry, curElemVolVars);
            if (!this->assembler().isStationaryProblem())
                this->prevElemVolVars().bindElement(element, fvGeometry, this->assembler().prevSol()[domainId]);
        }
        else
        {
            auto& prevElemVolVars = this->prevElemVolVars();
            const auto& prevSol = this->assembler().prevSol()[domainId];
 
            curElemVolVars.bindElement(element, fvGeometry, curSol);
            prevElemVolVars.bind(element, fvGeometry, prevSol);
            elemFluxVarsCache.bind(element, fvGeometry, prevElemVolVars);
        }
    }
 
    const Problem& problem() const
    { return this->assembler().problem(domainId); }
 
    CouplingManager& couplingManager()
    { return couplingManager_; }
 
private:
    CouplingManager& couplingManager_; 
};
 
template<std::size_t id, class TypeTag, class Assembler, DiffMethod DM = DiffMethod::numeric, bool implicit = true>
class SubDomainBoxLocalAssembler;
 
template<std::size_t id, class TypeTag, class Assembler>
class SubDomainBoxLocalAssembler<id, TypeTag, Assembler, DiffMethod::numeric, /*implicit=*/true>
: public SubDomainBoxLocalAssemblerBase<id, TypeTag, Assembler,
             SubDomainBoxLocalAssembler<id, TypeTag, Assembler, DiffMethod::numeric, true>, true >
{
    using ThisType = SubDomainBoxLocalAssembler<id, TypeTag, Assembler, DiffMethod::numeric, /*implicit=*/true>;
    using ParentType = SubDomainBoxLocalAssemblerBase<id, TypeTag, Assembler, ThisType, /*implicit=*/true>;
    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
    using VolumeVariables = GetPropType<TypeTag, Properties::VolumeVariables>;
    using ElementResidualVector = typename ParentType::LocalResidual::ElementResidualVector;
 
    using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView;
    using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
    using FVElementGeometry = typename GridGeometry::LocalView;
    using GridView = typename GridGeometry::GridView;
    using Element = typename GridView::template Codim<0>::Entity;
 
    enum { numEq = GetPropType<TypeTag, Properties::ModelTraits>::numEq() };
    enum { dim = GridView::dimension };
 
    static constexpr bool enableGridFluxVarsCache = getPropValue<TypeTag, Properties::EnableGridFluxVariablesCache>();
    static constexpr bool enableGridVolVarsCache = getPropValue<TypeTag, Properties::EnableGridVolumeVariablesCache>();
    static constexpr auto domainI = Dune::index_constant<id>();
 
public:
    using ParentType::ParentType;
 
    template<class JacobianMatrixDiagBlock, class GridVariables>
    ElementResidualVector assembleJacobianAndResidualImpl(JacobianMatrixDiagBlock& A, GridVariables& gridVariables)
    {
        // Calculate derivatives of all dofs in stencil with respect to the dofs in the element. In the //
        // neighboring elements we do so by computing the derivatives of the fluxes which depend on the //
        // actual element. In the actual element we evaluate the derivative of the entire residual.     //
 
        // get the vecor of the acutal element residuals
        const auto origResiduals = this->evalLocalResidual();
 
        // compute the derivatives of this element with respect to all of the element's dofs and add them to the Jacobian //
        const auto& element = this->element();
        const auto& fvGeometry = this->fvGeometry();
        const auto& curSol = this->curSol()[domainI];
        auto&& curElemVolVars = this->curElemVolVars();
 
        // create the element solution
        auto elemSol = elementSolution(element, curSol, fvGeometry.gridGeometry());
 
        auto partialDerivs = origResiduals;
        partialDerivs = 0.0;
 
        for (auto&& scv : scvs(fvGeometry))
        {
            // dof index and corresponding actual pri vars
            const auto dofIdx = scv.dofIndex();
            auto& curVolVars = this->getVolVarAccess(gridVariables.curGridVolVars(), curElemVolVars, scv);
            const VolumeVariables origVolVars(curVolVars);
 
            // calculate derivatives w.r.t to the privars at the dof at hand
            for (int pvIdx = 0; pvIdx < numEq; pvIdx++)
            {
                partialDerivs = 0.0;
 
                auto evalResiduals = [&](Scalar priVar)
                {
                    // update the volume variables and compute element residual
                    const auto localDofIndex = scv.localDofIndex();
                    elemSol[localDofIndex][pvIdx] = priVar;
                    curVolVars.update(elemSol, this->problem(), element, scv);
                    this->couplingManager().updateCouplingContext(domainI, *this, domainI, scv.dofIndex(), elemSol[localDofIndex], pvIdx);
                    return this->evalLocalResidual();
                };
 
                // derive the residuals numerically
                static const int numDiffMethod = getParamFromGroup<int>(this->problem().paramGroup(), "Assembly.NumericDifferenceMethod");
                static const NumericEpsilon<Scalar, numEq> eps_{this->problem().paramGroup()};
                NumericDifferentiation::partialDerivative(evalResiduals, elemSol[scv.localDofIndex()][pvIdx], partialDerivs, origResiduals,
                                                          eps_(elemSol[scv.localDofIndex()][pvIdx], pvIdx), numDiffMethod);
 
                // update the global stiffness matrix with the current partial derivatives
                for (auto&& scvJ : scvs(fvGeometry))
                {
                      for (int eqIdx = 0; eqIdx < numEq; eqIdx++)
                      {
                          // A[i][col][eqIdx][pvIdx] is the rate of change of
                          // the residual of equation 'eqIdx' at dof 'i'
                          // depending on the primary variable 'pvIdx' at dof
                          // 'col'.
                          A[scvJ.dofIndex()][dofIdx][eqIdx][pvIdx] += partialDerivs[scvJ.localDofIndex()][eqIdx];
                      }
                }
 
                // restore the original state of the scv's volume variables
                curVolVars = origVolVars;
 
                // restore the original element solution and coupling context
                elemSol[scv.localDofIndex()][pvIdx] = curSol[scv.dofIndex()][pvIdx];
                this->couplingManager().updateCouplingContext(domainI, *this, domainI, scv.dofIndex(), elemSol[scv.localDofIndex()], pvIdx);
            }
        }
 
        // evaluate additional derivatives that might arise from the coupling
        this->couplingManager().evalAdditionalDomainDerivatives(domainI, *this, origResiduals, A, gridVariables);
 
        return origResiduals;
    }
 
    template<std::size_t otherId, class JacobianBlock, class GridVariables>
    void assembleJacobianCoupling(Dune::index_constant<otherId> domainJ, JacobianBlock& A,
                                  const ElementResidualVector& res, GridVariables& gridVariables)
    {
        // Calculate derivatives of all dofs in the element with respect to all dofs in the coupling stencil. //
 
        // get some aliases for convenience
        const auto& element = this->element();
        const auto& fvGeometry = this->fvGeometry();
        auto&& curElemVolVars = this->curElemVolVars();
        auto&& elemFluxVarsCache = this->elemFluxVarsCache();
 
        // get element stencil informations
        const auto& stencil = this->couplingManager().couplingStencil(domainI, element, domainJ);
 
        // convenience lambda for call to update self
        auto updateCoupledVariables = [&] ()
        {
            // Update ourself after the context has been modified. Depending on the
            // type of caching, other objects might have to be updated. All ifs can be optimized away.
            if (enableGridFluxVarsCache)
            {
                if (enableGridVolVarsCache)
                    this->couplingManager().updateCoupledVariables(domainI, *this, gridVariables.curGridVolVars(), gridVariables.gridFluxVarsCache());
                else
                    this->couplingManager().updateCoupledVariables(domainI, *this, curElemVolVars, gridVariables.gridFluxVarsCache());
            }
            else
            {
                if (enableGridVolVarsCache)
                    this->couplingManager().updateCoupledVariables(domainI, *this, gridVariables.curGridVolVars(), elemFluxVarsCache);
                else
                    this->couplingManager().updateCoupledVariables(domainI, *this, curElemVolVars, elemFluxVarsCache);
            }
        };
 
        const auto& curSolJ = this->curSol()[domainJ];
        for (const auto globalJ : stencil)
        {
            // undeflected privars and privars to be deflected
            const auto origPriVarsJ = curSolJ[globalJ];
            auto priVarsJ = origPriVarsJ;
 
            // the undeflected coupling residual
            const auto origResidual = this->couplingManager().evalCouplingResidual(domainI, *this, domainJ, globalJ);
 
            for (int pvIdx = 0; pvIdx < JacobianBlock::block_type::cols; ++pvIdx)
            {
                auto evalCouplingResidual = [&](Scalar priVar)
                {
                    priVarsJ[pvIdx] = priVar;
                    this->couplingManager().updateCouplingContext(domainI, *this, domainJ, globalJ, priVarsJ, pvIdx);
                    updateCoupledVariables();
                    return this->couplingManager().evalCouplingResidual(domainI, *this, domainJ, globalJ);
                };
 
                // derive the residuals numerically
                ElementResidualVector partialDerivs(element.subEntities(dim));
 
                const auto& paramGroup = this->assembler().problem(domainJ).paramGroup();
                static const int numDiffMethod = getParamFromGroup<int>(paramGroup, "Assembly.NumericDifferenceMethod");
                static const auto epsCoupl = this->couplingManager().numericEpsilon(domainJ, paramGroup);
 
                NumericDifferentiation::partialDerivative(evalCouplingResidual, origPriVarsJ[pvIdx], partialDerivs, origResidual,
                                                          epsCoupl(origPriVarsJ[pvIdx], pvIdx), numDiffMethod);
 
                // update the global stiffness matrix with the current partial derivatives
                for (auto&& scv : scvs(fvGeometry))
                {
                    for (int eqIdx = 0; eqIdx < numEq; eqIdx++)
                    {
                        // A[i][col][eqIdx][pvIdx] is the rate of change of
                        // the residual of equation 'eqIdx' at dof 'i'
                        // depending on the primary variable 'pvIdx' at dof
                        // 'col'.
 
                        // If the dof is coupled by a Dirichlet condition,
                        // set the derived value only once (i.e. overwrite existing values).
                        // For other dofs, add the contribution of the partial derivative.
                        const auto bcTypes = this->elemBcTypes()[scv.localDofIndex()];
                        if (bcTypes.isCouplingDirichlet(eqIdx))
                            A[scv.dofIndex()][globalJ][eqIdx][pvIdx] = partialDerivs[scv.localDofIndex()][eqIdx];
                        else if (bcTypes.isDirichlet(eqIdx))
                            A[scv.dofIndex()][globalJ][eqIdx][pvIdx] = 0.0;
                        else
                            A[scv.dofIndex()][globalJ][eqIdx][pvIdx] += partialDerivs[scv.localDofIndex()][eqIdx];
                    }
                }
 
                // restore the current element solution
                priVarsJ[pvIdx] = origPriVarsJ[pvIdx];
 
                // restore the undeflected state of the coupling context
                this->couplingManager().updateCouplingContext(domainI, *this, domainJ, globalJ, priVarsJ, pvIdx);
            }
 
            // Restore original state of the flux vars cache and/or vol vars.
            // This has to be done in case they depend on variables of domainJ before
            // we continue with the numeric derivative w.r.t the next globalJ. Otherwise,
            // the next "origResidual" will be incorrect.
            updateCoupledVariables();
        }
    }
};
 
template<std::size_t id, class TypeTag, class Assembler>
class SubDomainBoxLocalAssembler<id, TypeTag, Assembler, DiffMethod::numeric, /*implicit=*/false>
: public SubDomainBoxLocalAssemblerBase<id, TypeTag, Assembler,
             SubDomainBoxLocalAssembler<id, TypeTag, Assembler, DiffMethod::numeric, false>, false >
{
    using ThisType = SubDomainBoxLocalAssembler<id, TypeTag, Assembler, DiffMethod::numeric, /*implicit=*/false>;
    using ParentType = SubDomainBoxLocalAssemblerBase<id, TypeTag, Assembler, ThisType, /*implicit=*/false>;
    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
    using VolumeVariables = GetPropType<TypeTag, Properties::VolumeVariables>;
    using ElementResidualVector = typename ParentType::LocalResidual::ElementResidualVector;
    using ElementVolumeVariables = typename GetPropType<TypeTag, Properties::GridVolumeVariables>::LocalView;
    using GridGeometry = GetPropType<TypeTag, Properties::GridGeometry>;
    using FVElementGeometry = typename GridGeometry::LocalView;
    using GridView = typename GridGeometry::GridView;
    using Element = typename GridView::template Codim<0>::Entity;
 
    enum { numEq = GetPropType<TypeTag, Properties::ModelTraits>::numEq() };
    enum { dim = GridView::dimension };
 
    static constexpr bool enableGridFluxVarsCache = getPropValue<TypeTag, Properties::EnableGridFluxVariablesCache>();
    static constexpr bool enableGridVolVarsCache = getPropValue<TypeTag, Properties::EnableGridVolumeVariablesCache>();
    static constexpr auto domainI = Dune::index_constant<id>();
 
public:
    using ParentType::ParentType;
 
    template<class JacobianMatrixDiagBlock, class GridVariables>
    ElementResidualVector assembleJacobianAndResidualImpl(JacobianMatrixDiagBlock& A, GridVariables& gridVariables)
    {
        if (this->assembler().isStationaryProblem())
            DUNE_THROW(Dune::InvalidStateException, "Using explicit jacobian assembler with stationary local residual");
 
        // get some aliases for convenience
        const auto& element = this->element();
        const auto& fvGeometry = this->fvGeometry();
        const auto& curSol = this->curSol()[domainI];
        auto&& curElemVolVars = this->curElemVolVars();
 
        // get the vector of the actual (undeflected) element residuals
        const auto origResiduals = this->evalLocalResidual();
        const auto origStorageResiduals = this->evalLocalStorageResidual();
 
        // Calculate derivatives of all dofs in stencil with respect to the dofs in the element. In the //
        // neighboring elements all derivatives are zero. For the assembled element only the storage    //
        // derivatives are non-zero.                                                                    //
 
        // create the element solution
        auto elemSol = elementSolution(element, curSol, fvGeometry.gridGeometry());
 
        // create the vector storing the partial derivatives
        ElementResidualVector partialDerivs(element.subEntities(dim));
 
        // calculation of the derivatives
        for (auto&& scv : scvs(fvGeometry))
        {
            // dof index and corresponding actual pri vars
            const auto dofIdx = scv.dofIndex();
            auto& curVolVars = this->getVolVarAccess(gridVariables.curGridVolVars(), curElemVolVars, scv);
            const VolumeVariables origVolVars(curVolVars);
 
            // calculate derivatives w.r.t to the privars at the dof at hand
            for (int pvIdx = 0; pvIdx < numEq; pvIdx++)
            {
                partialDerivs = 0.0;
 
                auto evalStorage = [&](Scalar priVar)
                {
                    // auto partialDerivsTmp = partialDerivs;
                    elemSol[scv.localDofIndex()][pvIdx] = priVar;
                    curVolVars.update(elemSol, this->problem(), element, scv);
                    return this->evalLocalStorageResidual();
                };
 
                // derive the residuals numerically
                static const NumericEpsilon<Scalar, numEq> eps_{this->problem().paramGroup()};
                static const int numDiffMethod = getParamFromGroup<int>(this->problem().paramGroup(), "Assembly.NumericDifferenceMethod");
                NumericDifferentiation::partialDerivative(evalStorage, elemSol[scv.localDofIndex()][pvIdx], partialDerivs, origStorageResiduals,
                                                          eps_(elemSol[scv.localDofIndex()][pvIdx], pvIdx), numDiffMethod);
 
                // update the global stiffness matrix with the current partial derivatives
                for (int eqIdx = 0; eqIdx < numEq; eqIdx++)
                {
                    // A[i][col][eqIdx][pvIdx] is the rate of change of
                    // the residual of equation 'eqIdx' at dof 'i'
                    // depending on the primary variable 'pvIdx' at dof
                    // 'col'.
                    A[dofIdx][dofIdx][eqIdx][pvIdx] += partialDerivs[scv.localDofIndex()][eqIdx];
                }
 
                // restore the original state of the scv's volume variables
                curVolVars = origVolVars;
 
                // restore the original element solution
                elemSol[scv.localDofIndex()][pvIdx] = curSol[scv.dofIndex()][pvIdx];
                // TODO additional dof dependencies
            }
        }
 
        // return the undeflected residual
        return origResiduals;
    }
 
    template<std::size_t otherId, class JacobianBlock, class GridVariables>
    void assembleJacobianCoupling(Dune::index_constant<otherId> domainJ, JacobianBlock& A,
                                  const ElementResidualVector& res, GridVariables& gridVariables)
    {}
};
 
} // end namespace Dumux
 
#endif