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We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0 for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the proposed HDG-P0 schemes, which hasn’t appeared in the literature yet for HDG discretizations as far as numerical integration is concerned. Moreover, we propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes. In both cases, we first establish the equivalence of the statically condensed HDG system with a (slightly modified) nonconforming Crouzeix–Raviart (CR) discretization, where the global (piecewise-constant) HDG finite element space on the mesh skeleton has a natural one-to-one correspondence to the nonconforming CR (piecewise-linear) finite element space that live on the whole mesh. This equivalence then allows us to use the well-established nonconforming geometry multigrid theory to precondition the condensed HDG system. Numerical results in two- and three-dimensions are presented to verify our theoretical findings.
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Heterogeneous Stefan problem and permafrost models with P0-P0 finite elements and fully implicit monolithic solver
We consider heat conduction models with phase change in heterogeneous materials. We are motivated by important applications including heat conduction in permafrost, phase change materials (PCM), and human tissue. We focus on the mathematical and computational challenges associated with the nonlinear and discontinuous character of constitutive relationships related to the presence of free boundaries and material interfaces. We propose a monolithic discretization framework based on lowest order mixed finite elements on rectangular grids well known for its conservative properties. We implement this scheme which we call P0-P0 as cell centered finite differences, and combine with a fully implicit time stepping scheme. We show that our algorithm is robust and compares well to piecewise linear approaches. While various basic theoretical properties of the algorithms are well known, we prove several results for the new heterogeneous framework, and point out challenges and open questions; these include the approximability of fluxes by piecewise continuous linears, while the true flux features a jump. We simulate a variety of scenarios of interest.
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- PAR ID:
- 10331390
- Date Published:
- Journal Name:
- Electronic Research Archive
- Volume:
- 30
- Issue:
- 4
- ISSN:
- 2688-1594
- Page Range / eLocation ID:
- 1477 to 1531
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3934/era.2022078
