template <class VectorType, class MatrixType, class FunctionType, size_t dim>
BackwardEuler<VectorType, MatrixType, FunctionType, dim>::BackwardEuler(
    MatrixType const &_A, MatrixType const &_M, MatrixType const &_C,
    double _wc, VectorType const &_u_initial, VectorType const &_v_initial,
    Dune::BitSetVector<dim> const &_dirichletNodes,
    FunctionType const &_dirichletFunction)
    : A(_A),
      M(_M),
      C(_C),
      wc(_wc),
      u(_u_initial),
      v(_v_initial),
      dirichletNodes(_dirichletNodes),
      dirichletFunction(_dirichletFunction) {}

template <class VectorType, class MatrixType, class FunctionType, size_t dim>
void BackwardEuler<VectorType, MatrixType, FunctionType, dim>::nextTimeStep() {
  v_o = v;
  u_o = u;
}

template <class VectorType, class MatrixType, class FunctionType, size_t dim>
void BackwardEuler<VectorType, MatrixType, FunctionType, dim>::setup(
    VectorType const &ell, double _tau, double relativeTime, VectorType &rhs,
    VectorType &iterate, MatrixType &AM) {
  postProcessCalled = false;

  tau = _tau;

  /* We start out with the formulation

       M a + C v + A u = ell

     Backward Euler means

       a1 = 1.0/tau ( v1 - v0 )
       u1 = tau v1 + u0

     in summary, we get at time t=1

       M [1.0/tau ( v1 - v0 )] + C v1
       + A [tau v1 + u0] = ell

     or

       1.0/tau M v1 + C v1 + tau A v1
       = [1.0/tau M + C + tau A] v1
       = ell + 1.0/tau M v0 - A u0
  */

  // set up LHS (for fixed tau, we'd only really have to do this once)
  {
    Dune::MatrixIndexSet indices(A.N(), A.M());
    indices.import(A);
    indices.import(M);
    indices.import(C);
    indices.exportIdx(AM);
  }
  AM = 0.0;
  Arithmetic::addProduct(AM, (1.0 - wc) / tau, M);
  Arithmetic::addProduct(AM, wc, C);
  Arithmetic::addProduct(AM, tau, A);

  // set up RHS
  {
    rhs = ell;
    Arithmetic::addProduct(rhs, (1.0 - wc) / tau, M, v_o);
    Arithmetic::subtractProduct(rhs, A, u_o);
  }

  // v_o makes a good initial iterate; we could use anything, though
  iterate = 0.0;

  for (size_t i = 0; i < dirichletNodes.size(); ++i)
    switch (dirichletNodes[i].count()) {
      case 0:
        continue;
      case dim:
        iterate[i] = 0;
        dirichletFunction.evaluate(relativeTime, iterate[i][0]);
        break;
      case 1:
        if (dirichletNodes[i][0]) {
          dirichletFunction.evaluate(relativeTime, iterate[i][0]);
          break;
        }
        if (dirichletNodes[i][1]) {
          iterate[i][1] = 0;
          break;
        }
        assert(false);
      default:
        assert(false);
    }
}

template <class VectorType, class MatrixType, class FunctionType, size_t dim>
void BackwardEuler<VectorType, MatrixType, FunctionType, dim>::postProcess(
    VectorType const &iterate) {
  postProcessCalled = true;

  v = iterate;

  u = u_o;
  Arithmetic::addProduct(u, tau, v);
}

template <class VectorType, class MatrixType, class FunctionType, size_t dim>
void BackwardEuler<VectorType, MatrixType, FunctionType,
                   dim>::extractDisplacement(VectorType &displacement) const {
  if (!postProcessCalled)
    DUNE_THROW(Dune::Exception, "It seems you forgot to call postProcess!");

  displacement = u;
}

template <class VectorType, class MatrixType, class FunctionType, size_t dim>
void BackwardEuler<VectorType, MatrixType, FunctionType, dim>::extractVelocity(
    VectorType &velocity) const {
  if (!postProcessCalled)
    DUNE_THROW(Dune::Exception, "It seems you forgot to call postProcess!");

  velocity = v;
}