HowTo: Create a custom preconditioner using only matrix actions

For many problems of interest the off the shelf preconditioners are insufficient and something more tailored to the equations of interest is required. MFEM has a flexible approach to defining preconditioners enabled by deriving from the existing Solver class and overriding the necessary methods to define the action. See the following example:

// Define a custom solver class that can be used as the preconditioner for a broader problem solvers
// Here we will define the example preconditioner:  P x = M x + Ainv x
class SumSolver : mfem::Solver
    const mfem::Operator *M;            //Since these are Operators only their
    const mfem::Operator *Ainv;         //actions need to be defined

    SumSolver(const mfem::Operator *M_, const mfem::Operator *Ainv_)
      : mfem::Solver(M_->Height(), M_->Width(), false)
        MFEM_VERIFY(M_->Height() == Ainv_->Height());
        MFEM_VERIFY(M_->Width() == Ainv_->Width());
        M = M_;
        Ainv = Ainv_;

    // Define the action of the Solver
    // y = P x =  M x + Ainv x
    void Mult(const mfem::Vector &x, mfem::Vector &y) const
      y = 0.0;
      mfem::Vector M_x(M->Height());
      mfem::Vector Ainv_x(Ainv->Height());
      M->Mult(x, M_x);                  // M_x = A x
      Ainv->Mult(x, Ainv_x);    // Ainv_x = Ainv x
      y.Add(1.0, M_x);                  // y += M_x
      y.Add(1.0, Ainv_x);               // y += Ainv_x

    void SetOperator(const Operator &op) { M = &op;};

In this example we defined a new MFEM solver that can be applied as a preconditioner for a broader solution. In this case we demonstrated an example where we have a matrix M, the action of the inverse of a matrix A, and we want to define the action of a preconditioner that is the sum of the two. In this case we cannot simply sum the matrices to form the new preconditioner because we don't have access to the elements of Ainv. As you can see this approach is quite flexible and can be utilized to create custom preconditioners of arbitrary complexity.