The Eigenvalue Problem for the 2D Laplacian in ℋ-Matrix Arithmetic and Application to the Heat and Wave Equation

Abstract

A class of matrices (ℋ-matrices) has recently been introduced by Hackbusch for approximating large and fully populated matrices arising from FEM and BEM applications. These matrices are data-sparse and allow approximate matrix operations of almost linear complexity. In the present paper, we choose a special class of ℋ-matrices that provides a good approximation to the inverse of the discrete 2D Laplacian. For these 2D ℋ-matrices we study the blockwise recursive schemes for block triangular linear systems of equations and the Cholesky and LDL^T factorization in an approximate arithmetic of almost linear complexity. Using the LDL^T factorization we compute eigenpairs of the discrete 2D Laplacian in ℋ-matrix arithmetic by means of a so-called simultaneous iteration for computing invariant subspaces of non-Hermitian matrices due to Stewart. We apply the ℋ-matrix techniques to approximate the solutions of the high-frequency 2D wave equation for smooth initial data and the 2D heat equation for arbitrary initial data by spectral decomposition of the discrete 2D Laplacian in, up to logarithmic factors, optimal complexity.