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Int. J. Numer. Anal. Mod., 16 (2019), pp. 891-924.
Published online: 2019-08
[An open-access article; the PDF is free to any online user.]
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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.
}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13259.html} }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.