Local projection stabilization for convection–diffusion–reaction equations on surfaces

Abstract

The numerical solution of convection–diffusion–reaction equations in two and three dimensional domains Ω is thoroughly studied and well understood. Stabilized finite element methods have been developed to handle boundary or interior layers and to localize and suppress unphysical oscillations. Much less is known on convection–diffusion–reaction equations on surfaces Γ=∂Ω. We propose a fitted finite element local projection stabilization (LPS) for convection–diffusion–reaction equations on surfaces based on a linear surface approximation and continuous, piecewise linear finite elements. Unique solvability of the continuous and discrete problem is established. A-priori error estimates show the O(h3∕2) convergence in mesh-dependent norms for smooth solutions in the convection-dominated case. The associated bilinear form satisfies an inf–sup condition in a norm stronger than the standard LPS-norm leading to improved stability properties. Numerical test examples demonstrate the properties and the potential of the proposed method.