Year: 2019
Author: David Wells, Jeffrey Banks
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 6 : pp. 891–924
Abstract
Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel approach to calculating accurate approximations of boundary derivatives of elliptic problems. We describe a new continuous finite element method based on $p$-refinement of cells adjacent to the boundary that increases the local degree of the approximation. We prove that the order of the approximation on the $p$-refined cells is, in 1D, determined by the rate of convergence at the mesh vertex connecting the higher and lower degree cells and that this approach can be extended, in a restricted setting, to 2D problems. The proven convergence rates are numerically verified by a series of experiments in both 1D and 2D. Finally, we demonstrate, with additional numerical experiments, that the $p$-refinement method works in more general geometries.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2019-IJNAM-13259
International Journal of Numerical Analysis and Modeling, Vol. 16 (2019), Iss. 6 : pp. 891–924
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 34
Keywords: Finite elements superconvergence elliptic equations numerical analysis scientific computing.