A Reconstructed-Laplacian Method for the Surface Biharmonic Equation on Parametric Meshes

Abstract

We develop and analyze a continuous/discontinuous Galerkin (CDG) method based on reconstructed surface Laplacians for the biharmonic equation on a smooth closed surface. Continuous mapped finite elements of degree $k\ge 2$ are used on fitted parametric meshes of degree $r\ge 1$, while a discontinuous degree-$(k-2)$ lifting corrects the broken Laplace-Beltrami operator for two-sided conormal-flux jumps. The resulting completed-square form is coercive on the mean-zero space for every fixed $\beta >0$, without requiring a sufficiently large penalty parameter. Under the standard geometric assumptions, we prove that the energy and reconstructed-Laplacian errors are $\mathcal{O}(h^{k-1}+h^r)$, and the $L^2$-error is $\mathcal{O}(h^{q_k}+h^{r+1})$, where $q_2=2$ and $q_k=k+1$ for $k\ge 3$. Benchmark computations support these rates, while a surface Swift-Hohenberg experiment illustrates the extension of the method to nonlinear Laplacian-dominated models.

30 pages