Preserving Superconvergence of Spectral Elements for Curved Domains via $$h$$ and $$p$$-Geometric Refinement

Technical Report · Tue Nov 14 04:00:00 EST 2023

DOI:https://doi.org/10.2172/2208785· OSTI ID:2208785

Jones, Jacob Taylor ^[1]

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1. Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Stony Brook University, NY (United States)

Spectral element methods (SEM), extensions of finite element methods (FEM), have emerged as significant techniques for solving partial differential equations in physics and engineering. SEM can potentially deliver superior accuracy due to the potential superconvergence in nodal solutions for well-shaped tensor-product elements. However, the accuracy of SEM often degrades in complex geometries due to geometric inaccuracies near curved boundaries and the loss of superconvergence with simplicial or non-tensor-product elements. To overcome the first issue, we propose using h- and p-geometric refinement, which refines the mesh near high-curvature regions and increases the degree of geometric basis functions, respectively. We show that when using mixed-element meshes with tensor-product elements in the interior of the domain, curvature-based geometric refinement near boundaries can improve the accuracy of the interior elements by reducing pollution errors and preserving the superconvergence in nodal solutions. To address the second issue, we introduce a post-processing technique using the adaptive extended stencil finite element method (AES-FEM) to recover the accuracy near the curved boundaries. The combination of curvature-based geometric refinement and accurate post-processing offers an effective and easier-to-implement alternative to methods reliant on exact geometries. We demonstrate our techniques by solving the convection-diffusion equation in 2D and show up to two orders of magnitude of improvement in the solution accuracy, even when the elements are poorly shaped near boundaries.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• DOE Contract Number: 89233218CNA000001
• OSTI ID: 2208785
• Report Number(s): LA-UR--23-32902
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
curved boundaries
geometric refinement
spectral element methods
superconvergence