Adaptive $\mathcal{H}$-Matrix Computations in Linear Elasticity

TL;DR

This paper introduces adaptive hierarchical matrix techniques for efficiently solving boundary integral equations in linear elasticity, significantly reducing computational costs through adaptive algorithms and error estimation.

Contribution

It presents novel adaptive algorithms based on hierarchical matrices and cross approximation for boundary element methods in linear elasticity.

Findings

Efficient adaptive matrix-vector multiplication scheme developed.

Error estimators guide adaptivity for matrix approximation.

Reduced computational complexity in solving elasticity problems.

Abstract

This article deals with the adaptive and approximative computation of the Lam\'e equations. The equations of linear elasticity are considered as boundary integral equations and solved in the setting of the boundary element method (BEM). Using BEM, one is faced with the solution of a system of equations with a fully populated system matrix, which is in general very costly. Some adaptive algorithms based on hierarchical matrices and the adaptive cross approximation are proposed. At first, an adaptive matrix-vector multiplication scheme is introduced for the efficient treatment of multiplying discretizations with given data. The strategy, to reach this aim, is to use error estimators and techniques known from adaptivity. The case of approximating the system matrix appearing in the linear system of equations with this new type of adaptivity is also discussed.