Superconvergence And Mesh Generation And Optimization For Finite Element Methods Of Interface Problem And Laplace-Beltrami Problem

In this study, we focus on the superconvergence phenomenon in2D elliptic interface problem and Laplace-Beltrami problem.For2D elliptic interface problem, adaptive mesh refinement and the Borgers’algo-rithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolant of the exact solution. Based on this superconvergence result, a maximal norm error estimate is obtained. The basic idea of the Borgers’algorithm is to use a Cartesian grid but perturb only the grid points near the interface onto the interface and choose an appropriate diagonal of perturbed quadrilaterals to fit the interface and maintain the mesh quality. The final body-fitted mesh is shape regular and topologically equivalent to the Cartesian grid. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients.For Laplace-Beltrami problem, superconvergence results and several gradient re-covery methods of finite element methods in flat spaces are generalized to the surface linear finite element method for the Laplace-Beltrami equation on general surfaces with mildly structured triangular meshes. For a large class of practically useful grids, the sur-face linear finite element solution is proven to be superclose to an interpolant of the exact solution of the Laplace-Beltrami equation, and as a result various postprocessing gradient recovery, including simple and weighted averaging, local and global L2-projections, and Zienkiewicz and Zhu (ZZ) schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution. Numerical experi-ments are presented to confirm the theoretical results. We also design a smooth density function which includes the surface curvature information and can be used in the CVT optimization of surface triangulation. The numerical tests show that the density function is effective to improve the quality of the surface triangulation.