Title: Curved meshes for a diffusion problem with a surface Laplacian
Abstract
We study a diffusion problem with boundary conditions that involve a surface Laplacian. We present the numerical analysis of a high-order finite element discretization. A smooth domain is considered for the definition of this boundary operator. The discrtized domain however presents a geometrical error. We use curved meshes to reduce this error. Moreover, a lift operator is introduced to compare the discrete solution on the discrete domain with the exact solution on the exact domain. A priori error bounds are established, involving separate contributions of a geometric error and an approximation error. Numerical 2D and 3D experiments validate the theoretical results, emphasizing the optimality of the a priori bounds. A super-convergence phenomenon appears in the presence of quadratic meshes.
