TY - JOUR
T1 - A $C^1$-Conforming Gauss Collocation Method for Elliptic Equations and Superconvergence Analysis Over Rectangular Meshes
AU - Cao , Waixiang
AU - Jia , Lueling
AU - Zhang , Zhimin
JO - CSIAM Transactions on Applied Mathematics
VL - 2
SP - 320
EP - 349
PY - 2024
DA - 2024/05
SN - 5
DO - http://doi.org/10.4208/csiam-am.SO-2022-0018
UR - https://global-sci.org/intro/article_detail/csiam-am/23124.html
KW - Hermite interpolation, $C^1$-conforming, superconvergence, Gauss collocation methods, Jacobi polynomials.
AB - <p style="text-align: justify;">This paper is concerned with a $C^1$-conforming Gauss collocation approximation to the solution of a model two-dimensional elliptic boundary problem. Superconvergence phenomena for the numerical solution at mesh nodes, at roots of a special Jacobi polynomial, and at the Lobatto and Gauss lines are identified with rigorous mathematical proof, when tensor products of $C^1$ piecewise polynomials of degree
not more than $k,$ $k≥3$ are used. This method is shown to be superconvergent with $(2k−2)$-th order accuracy in both the function value and its gradient at mesh nodes, $(k+2)$-th order accuracy at all interior roots of a special Jacobi polynomial, $(k+1)$-th
order accuracy in the gradient along the Lobatto lines, and $k$-th order accuracy in the
second-order derivative along the Gauss lines. Numerical experiments are presented
to indicate that all the superconvergence rates are sharp.</p>