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Numerical integration of the stiffness matrix in higher-order finite element (FE) methods is recognized as one of the heaviest computational tasks in an FE solver. The problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology. Making use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high-order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron. The key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions. Consistent with existing results, the presented algorithms for the integration of H1, H(curl), H(div), and L2 inner products, have the O(p7) computational complexity in contrast to the O(p9) cost of conventional integration routines. Use of Legendre polynomials for shape functions is critical in this implementation. Three boundary value problems under different variational formulations, requiring combinations of H1, H(div) and H(curl) test shape functions, were chosen to experimentally assess the computation time for constructing DPG element matrices, showing good correspondence with the expected rates.
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Evaluation of Inner Products of Implicitly Defined Finite Element Functions on Multiply Connected Planar Mesh Cells
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 \(H^1\) semi-inner product and \(L^2\) 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 \(L^2\) inner product. Several numerical examples demonstrate the high-order accuracy of this approach. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at both https://github.com/samreynoldsmath/PuncturedFEM and the supplementary materials (PuncturedFEM\_v0\_2\_5.zip [1.75MB]).
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- PAR ID:
- 10494566
- Publisher / Repository:
- Society for Industrial and Applied Mathematics
- Date Published:
- Journal Name:
- SIAM Journal on Scientific Computing
- Volume:
- 46
- Issue:
- 1
- ISSN:
- 1064-8275
- Page Range / eLocation ID:
- A338 to A359
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1137/23M1569332
