Published In
Siam Journal on Scientific Computing
Document Type
Pre-Print
Publication Date
6-16-2024
Subjects
Mathematics
Abstract
In recent years, there has been significant interest in the development of finite element methods defined on meshes that include rather general polytopes and curvilinear polygons. In the present work, we provide tools necessary to employ multiply connected mesh cells in planar domains, i.e., cells with holes, in finite element computations. Our focus is efficient evaluation of the 𝐻1𝐻1 semi-inner product and 𝐿2𝐿2 inner product of implicitly defined finite element functions of the types arising in boundary element based finite element methods and virtual element methods. Such functions are defined as solutions of Poisson problems having a polynomial source term and continuous boundary data. We show that the integrals of interest can be reduced to integrals along the boundaries of mesh cells, thereby avoiding the need to perform any computations in cell interiors. The dominating cost of this reduction is solving a relatively small Nyström system to obtain a Dirichlet-to-Neumann map, as well as the solution of two more Nyström systems to obtain an “anti-Laplacian” of a harmonic function, which is used for computing the 𝐿2𝐿2 inner product. Several numerical examples demonstrate the high-order accuracy of this approach.
Rights
© Copyright the author(s) 2024
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DOI
10.1137/23M1569332
Persistent Identifier
https://archives.pdx.edu/ds/psu/42042
Citation Details
Published as: Ovall, J. S., & Reynolds, S. E. (2024). Evaluation of Inner Products of Implicitly Defined Finite Element Functions on Multiply Connected Planar Mesh Cells. SIAM Journal on Scientific Computing, 46(1), A338–A359. https://doi.org/10.1137/23m1569332
Description
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published as: Evaluation of Inner Products of Implicitly Defined Finite Element Functions on Multiply Connected Planar Mesh Cells. SIAM Journal on Scientific Computing, 46(1), A338-A359.