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Mathematics of Computation

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Smoothed projections and mixed boundary conditions

Author: Martin W. Licht
Journal: Math. Comp. 88 (2019), 607-635
MSC (2010): Primary 65N30; Secondary 58A12
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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Additional Information

Martin W. Licht
Affiliation: Department of Mathematics, University of California San Diego, 9500 Gilman Drive MC0112, La Jolla, California 92093-0112
MR Author ID: 1225084

Keywords: Finite element exterior calculus, Hodge Laplace equation, smoothed projection, partial boundary conditions, mixed boundary conditions
Received by editor(s): November 6, 2016
Received by editor(s) in revised form: May 22, 2017, and January 1, 2017
Published electronically: April 10, 2018
Additional Notes: This research was supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, project 278011 STUCCOFIELDS
Article copyright: © Copyright 2018 American Mathematical Society