A simple and efficient interface-fitted mesh algorithm which can produce a semi-unstructured interface-fitted mesh in two and three dimensions quickly is developed in this paper. Elements in such interface-fitted meshes are not restricted to simplices but can be polygons or polyhedra. Especially in 3D, the polyhedra instead of tetrahedra can avoid slivers. Virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions. Algebraic multigrid solvers are used to solve the resulting linear algebraic system. Numerical results are presented to illustrate the effectiveness of our method.

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Superconvergence results and several gradient recovery 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 surface 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 post-processing gradient recovery, including simple and weighted averaging, local and global L 2-projections, and Z-Z schemes are devised and proven to be a better approximation of the true gradient than the gradient of the finite element solution. Numerical experiments are presented to confirm the theoretical results.