A layers capturing type H-adaptive finite element method for convection–diffusion–reaction equations on surfaces

Abstract

In this work, we study the finite element approximation of surface convection–diffusion–reaction equations which are important fundamental model problems in simulations of complex physical phenomena on moving interfaces, ultra-thin materials and biological films. The optimal error estimates of finite element method are shown. According to the analyzed results, the solutions of convection-dominated diffusion problems have low accuracy in the regions of layers. To overcome such drawback which may lead to non-physical oscillations, the adaptive mesh refinement method is considered. Due to the numerical characteristic of non-physical oscillations and layers, we present an error estimator which specializes in capturing the non-physical oscillations and layers. The mesh refinement marking strategies which take into account the large curvatures of the surfaces are developed. The proposed method can efficiently capture layers, non-physical oscillations and provide high resolution solutions with fewer degree of freedoms. A series of numerical examples are designed to demonstrate the effectiveness of the proposed H-adaptive method and make exploration of the solution behavior of convection-dominated diffusion problems on surfaces.