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 - <p style="text-align: justify;">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.</p>