Optimal preconditioning for Raviart-Thomas mixed formulation of second-order elliptic problems

We evaluate two preconditioning strategies for the indefinite linear system obtained from Raviart-Thomas mixed finite element formulation of a second-order elliptic problem with variable diffusion coefficients. It is known that the underlying saddle-point problem is well-posed in two function spaces, H(div) × L2and L2× H1, leading to the possibility of two distinct types of preconditioner. For homogeneous Dirichlet boundary conditions, the discrete problems are identical. This motivates our use of Raviart-Thomas approximation in both frameworks, yielding a nonconforming method in the second case. The focus is on linear algebra; we establish the optimality of two parameter-free block-diagonal preconditioners using basic properties of the finite element matrices. Uniform eigenvalue bounds are established and the impact of the PDE coefficients is explored in numerical experiments. A practical scheme is discussed, the key building block for which is a fast solver for a scalar diffusion operator based on algebraic multigrid. Trials of preconditioned MINRES illustrate that both preconditioning schemes are optimal with respect to the discretization parameter and robust with respect to the PDE coefficients.