Smoothed projections and mixed boundary conditions

Licht, Martin

doi:10.1090/mcom/3330

Smoothed projections and mixed boundary conditions

Author: Martin W. Licht
Journal: Math. Comp. 88 (2019), 607-635
MSC (2010): Primary 65N30; Secondary 58A12
DOI: https://doi.org/10.1090/mcom/3330
Published electronically: April 10, 2018
MathSciNet review: 3882278
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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.

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