A Condensed Constrained Nonconforming Mortar-based Approach for Preconditioning Finite Element Discretization Problems

TL;DR

This paper introduces a novel preconditioning approach for finite element problems using a modified mortar method that allows for efficient element-wise assembly and improved solver performance.

Contribution

It develops a new constrained nonconforming mortar-based reformulation enabling auxiliary space preconditioners with element-wise assembly for finite element discretizations.

Findings

Preconditioners improve convergence for high-order finite element methods.

The approach is effective on second order scalar elliptic problems.

Numerical results demonstrate enhanced solver efficiency.

Abstract

This paper presents and studies an approach for constructing auxiliary space preconditioners for finite element problems using a constrained nonconforming reformulation, that is based on a proposed modified version of the mortar method. The well-known mortar finite element discretization method is modified to admit a local structure, providing an element-by-element or subdomain-by-subdomain assembly property. This is achieved via the introduction of additional trace finite element spaces and degrees of freedom (unknowns) associated with the interfaces between adjacent elements or subdomains. The resulting nonconforming formulation and a reduced via static condensation Schur complement form on the interfaces are used in the construction of auxiliary space preconditioners for a given conforming finite element discretization problem. The properties of these preconditioners are studied and their performance is illustrated on model second order scalar elliptic problems utilizing high order elements.