On the algebraic construction of sparse multilevel approximations of elliptic tensor product problems

TL;DR

This paper introduces an algebraic multigrid-based approach to construct sparse multilevel approximations for elliptic tensor product problems, enabling application to complex geometries and unstructured grids.

Contribution

It presents a novel algebraic hierarchy construction method for sparse approximations, extending applicability to complex geometries and unstructured discretizations.

Findings

Algebraic hierarchy matches geometric convergence rates.

Applicable to black-box PDE solvers.

Enables sparse grid techniques on complex domains.

Abstract

We consider the solution of elliptic problems on the tensor product of two physical domains as e.g. present in the approximation of the solution covariance of elliptic partial differential equations with random input. Previous sparse approximation approaches used a geometrically constructed multilevel hierarchy. Instead, we construct this hierarchy for a given discretized problem by means of the algebraic multigrid method (AMG). Thereby, we are able to apply the sparse grid combination technique to problems given on complex geometries and for discretizations arising from unstructured grids, which was not feasible before. Numerical results show that our algebraic construction exhibits the same convergence behaviour as the geometric construction, while being applicable even in black-box type PDE solvers.