Khamron Mekchay

Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs

We prove convergence of adaptive finite element methods (AFEMs) for general (nonsymmetric) second order linear elliptic PDEs, thereby extending the result of Morin, Nochetto, and Siebert [{\it SIAM J. Numer.\ Anal.}, 38 (2000), pp. 466--488; {\it SIAM Rev.}, 44 (2002), pp. 631--658]. The proof relies on quasi-orthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEMs are a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and {both coercive and noncoercive} convection-diffusion PDE, illustrate the theory and yield optimal meshes.

AFEM for the Laplace-Beltrami operator on graphs: Design and conditional contraction property

We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on C 1 graphs in R d (d 2). We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in H 1 () and the surface error in W 1 1() due to approximation of . We devise a marking strategy to reduce the total error estimator, namely a suitably scaled sum of the energy, geometric, and inconsistency error estimators. We prove a conditional contraction property for the sum of the energy error and the total estimator; the conditional statement encodes resolution of in W 1 1. We conclude with one numerical experiment that illustrates the theory.

High-Order AFEM for the Laplace---Beltrami Operator: Convergence Rates

We present a new AFEM for the Laplace–Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally

Consistent weighted average flux of well-balanced TVD-RK discontinuous Galerkin method for shallow water flows

W^1_\infty

Dynamically Adaptive Tree Grid Modeling of Flood Inundation Based on Shallow Water Equations

W∞1 and piecewise in a suitable Besov class embedded in

Convergence of Adaptive Finite Element Methods

C^{1,\alpha }