Approximating inverse FEM matrices on non-uniform meshes with $${\mathcal{H}}$$-matrices

Abstract

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $${\mathcal{H}}$$ H -matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the $${\mathcal{H}}$$ H -matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

Exponential meshes and $\mathcal{H}$-matrices

“…A fully discrete approach, which avoids the final projection steps and leads to exponential convergence in the block rank, was taken in [FMP15,FMP21] in a FEM setting on quasi-uniform meshes and in the boundary element method (BEM) in [FMP16,FMP17,FMP20]. The generalization of [FMP15] to non-uniform meshes was achieved in [AFM21a] for low order FEM on certain classes of meshes that includes algebraically graded meshes. In the present work, we generalize [AFM21a] in several directions: first, we admit a larger class of meshes that includes certain shape-regular meshes that are graded exponentially towards a lower-dimensional manifold.…”

“…The generalization of [FMP15] to non-uniform meshes was achieved in [AFM21a] for low order FEM on certain classes of meshes that includes algebraically graded meshes. In the present work, we generalize [AFM21a] in several directions: first, we admit a larger class of meshes that includes certain shape-regular meshes that are graded exponentially towards a lower-dimensional manifold. In particular, we can show exponential approximability in the block rank for the inverses of FEM matrices arising in variants of the boundary concentrated FEM, [KM03].…”

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In [AFM21a], we proved that the inverse of the stiffness matrix of an h-version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block rank by H-matrices. Here, we improve on this result in multiple ways: (1) The class of meshes is significantly enlarged and includes certain exponentially graded meshes.

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$\mathcal{H}$-matrix approximability of inverses of FEM matrices for the time-harmonic Maxwell equations

“…While stiffness matrices arising from differential operators are sparse and are thus easily represented exactly in the standard H-matrix formats, the situation is more involved for the inverse. For FEM discretizations of various scalar elliptic operators, it was shown in [BH03,Beb07,FMP15,AFM20] that the inverse of the FEM-matrix can be approximated at a root exponential rate in the block rank. The works [FMP16, FMP17] show similar results for the boundary element method (BEM) and [FMP20] in a FEM-BEM coupling setting.…”

$\mathcal{H}$-inverses for RBF interpolation

“…Yet, a basic question remains whether the target, i.e., the inverse of the system matrix or the LU-factorization, can be represented in the H-matrix format. This question has been answered for finite element discretizations of elliptic PDEs in [BH03], recently improved in [FMP15] (to arbitrary accuracy) and [AFM21] (locally refined meshes), as well as for boundary element discretizations, [FMP16,FMP17], and the coupling of finite elements and boundary elements, [FMP20].…”