Auxiliary Space Preconditioning of Finite Element Equations Using a Nonconforming Interior Penalty Reformulation and Static Condensation

TL;DR

This paper introduces a novel auxiliary space preconditioning technique for finite element equations by reformulating interior penalty methods with additional interface unknowns, enabling efficient element-wise assembly and improved solver performance.

Contribution

It presents a new reformulation of interior penalty finite element methods that facilitates element-by-element assembly and develops auxiliary space preconditioners based on this reformulation.

Findings

Preconditioners show improved convergence on high-order elliptic problems.

The method enables efficient assembly and solution of finite element systems.

Performance is validated through numerical experiments on model problems.

Abstract

We modify the well-known interior penalty finite element discretization method so that it allows for element-by-element assembly. This is possible due to the introduction of additional unknowns associated with the interfaces between neighboring elements. The resulting bilinear form, and a Schur complement (reduced) version of it, are utilized in a number of auxiliary space preconditioners for the original conforming finite element discretization problem. These preconditioners are analyzed on the fine scale and their performance is illustrated on model second order scalar elliptic problems discretized with high order elements.