H-matrix approximability of the inverses of FEM matrices

Abstract

We study the question of approximability for the inverse of the FEM stiffness matrix for (scalar) second order elliptic boundary value problems by blockwise low rank matrices such as those given by the H-matrix format. We show that exponential convergence in the local block rank can be achieved. We also show that exponentially accurate LU-decompositions in the H-matrix format are possible for the stiffness matrices arising in the FEM. Unlike prior works, our analysis avoids any coupling of the block rank r and the mesh width h and also covers mixed Dirichlet-Neumann-Robin boundary conditions.

23 pages, 6 figures

References in corpus (6)

• A fast direct solver for structured linear systems by recursive skeletonization
• H-matrix approximability of the inverses of FEM matrices
• Existence of $\mathcal{H}$-matrix approximants to the inverses of BEM matrices: the simple-layer operator
• Existence of $\mathcal{H}$-matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator
• Hierarchical interpolative factorization for elliptic operators: differential equations
• An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads

Cited by in corpus (10)

• H-matrix approximability of the inverses of FEM matrices
• $\mathcal{H}$-matrix approximability of inverses of discretizations of the fractional Laplacian
• Existence of $\mathcal{H}$-matrix approximants to the inverse of BEM matrices: the hyper-singular integral operator
• Computing the eigenvalues of symmetric H2-matrices by slicing the spectrum
• Caccioppoli-type estimates and $\mathcal{H}$-Matrix approximations to inverses for FEM-BEM couplings
• Exponential meshes and $\mathcal{H}$-matrices
• $\mathcal{H}$-inverses for RBF interpolation
• Approximating inverse FEM matrices on non-uniform meshes with $\mathcal{H}$-matrices
• A generic multiresolution preconditioner for sparse symmetric systems
• $\mathcal{H}$-matrix based second moment analysis for rough random fields and finite element discretizations