Domain Decomposition Preconditioners for Mixed Finite-Element Discretization of High-Contrast Elliptic Problems

Abstract

In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale $$H(\mathrm {div})$$ problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this $$H(\mathrm {div})$$ problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart–Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale $$H(\mathrm {div})$$ system is bounded by $$C(1+\frac{H^2}{{\hat{\delta }}^2})(1+\log ^4(\frac{H}{h}))$$ , where $$\hat{\delta }$$ denotes the width of overlapping region, and H, h are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.