Cascadic Multilevel Algorithms for Symmetric Saddle Point Systems

TL;DR

This paper presents a multilevel cascadic algorithm for symmetric saddle point systems that efficiently combines stable finite element pairs and fast solvers, achieving near-optimal approximation rates.

Contribution

It introduces a new cascade-based multilevel algorithm with novel error estimates and a level change criterion for symmetric saddle point systems, applicable to various boundary value problems.

Findings

Achieves near-optimal approximation rates.

Efficiently combines finite element pairs and solvers.

Applicable to a broad class of boundary value problems.

Abstract

In this paper, we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems. The algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for it symmetric positive definite systems. On each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a Schur complement is implemented. The level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error. We prove new estimates that relate the iteration error and the residual for the constraint equation. The new estimates are the key ingredients in imposing an efficient level change criterion. The first iteration on each new level uses information about the best approximation of the discrete solution from the previous level. The theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non-increasing number of iterations on each level. Even though numerical results supporting the efficiency of the algorithms are presented for the Stokes system, the algorithms can be applied to a large class of boundary value problems, including first order systems that can be reformulated at the continuous level as symmetric saddle point problems, such as the Maxwell equations.