Volume 16, Issue 6
Using $p$-Refinement to Increase Boundary Derivative Convergence Rates

Int. J. Numer. Anal. Mod., 16 (2019), pp. 891-924.

Published online: 2019-08

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• 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.

• Keywords

Finite elements, superconvergence, elliptic equations, numerical analysis, scientific computing.

65N15, 65N30

wellsd2@rpi.edu (David Wells)

banksj3@rpi.edu (Jeffrey Banks)

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@Article{IJNAM-16-891, author = {Wells , David and Banks , Jeffrey}, title = {Using $p$-Refinement to Increase Boundary Derivative Convergence Rates}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2019}, volume = {16}, number = {6}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13259.html} }
TY - JOUR T1 - Using $p$-Refinement to Increase Boundary Derivative Convergence Rates AU - Wells , David AU - Banks , Jeffrey JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 891 EP - 924 PY - 2019 DA - 2019/08 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13259.html KW - Finite elements, superconvergence, elliptic equations, numerical analysis, scientific computing. AB -

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.

David Wells & Jeffrey Banks. (2019). Using $p$-Refinement to Increase Boundary Derivative Convergence Rates. International Journal of Numerical Analysis and Modeling. 16 (6). 891-924. doi:
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