Smoothed Projections and Mixed Boundary Conditions

TL;DR

This paper develops smoothed projection operators for finite element exterior calculus with mixed boundary conditions, ensuring stability and convergence of mixed finite element methods for the Hodge Laplace equation.

Contribution

It introduces new smoothed projections that commute with the exterior derivative, preserve boundary conditions, and are uniformly bounded, advancing finite element analysis with mixed boundary conditions.

Findings

Proved stability and quasi-optimal convergence of mixed finite element methods.

Constructed projections that commute with the exterior derivative and preserve boundary conditions.

Established density of smooth forms in Sobolev spaces over weakly Lipschitz domains.

Abstract

Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve homogeneous boundary conditions along a fixed boundary part, and satisfy uniform bounds for shape-regular families of triangulations and bounded polynomial degree. The existence of such projections implies stability and quasi-optimal convergence of mixed finite element methods for the Hodge Laplace equation with mixed boundary conditions. In addition, we prove the density of smooth differential forms in Sobolev spaces of differential forms over weakly Lipschitz domains with partial boundary conditions.