L2 AND POINTWISE A POSTERIORI ERROR ESTIMATES FOR FEM FOR ELLIPTIC PDE ON SURFACES

Some applications where finding the solution of partial differential equations on surfaces is of interest include: crystal growth, fluid mechanics and computer graphics. Surface FEM are widely used to solve such equations. A posteriori error estimators are computable measures of the FEM error in a given norm, and they are used to implement adaptive mesh refinement. In this work we derive a posteriori L₂ and pointwise error estimate for FEM for elliptic PDE's on surfaces. In particular we consider the model problem Δ_Γ u=f on Γ, where Δ_Γ is the Laplace Beltrami operator and Γ is a C³ surface without a boundary. Previous works include a paper by Demlow and Dziuk in 2007 where they proved a posteriori estimates for the energy norm. The a posteriori estimators for these methods have Galerkin and geometric components. We make use of standard approximation theory techniques to prove the reliability of the L₂ estimator; Green's functions properties are used for the point wise estimator. Our work includes numerical estimation of the dependence of the error bounds on the geometric properties of the surface. We provide also numerical experiments where the estimators have been used to implement an adaptive Finite Element Method over surfaces with different curvatures. One important conclusion is that as we let the curvature of the surface vary the geometric component of the error estimator becomes more important.