The Bulk-Surface Virtual Element Method for Reaction-Diffusion PDEs: Analysis and Applications (Communications in Computational Physics, 2023)
Bulk-surface partial differential equations (BS-PDEs) are prevalent in manyapplications such as cellular, developmental and plant biology as well as in engineering and material sciences. Novel numerical methods for BS-PDEs in three space dimensions (3D) are sparse. In this work, we present a bulk-surface virtual elementmethod (BS-VEM) for bulk-surface reaction-diffusion systems, a form of semilinearparabolic BS-PDEs in 3D. Unlike previous studies in two space dimensions (2D), the3D bulk is approximated with general polyhedra, whose outer faces constitute a flatpolygonal approximation of the surface. For this reason, the method is restricted tothe lowest order case where the geometric error is not dominant. The BS-VEM guarantees all the advantages of polyhedral methods such as easy mesh generation andfast matrix assembly on general geometries. Such advantages are much more relevantthan in 2D. Despite allowing for general polyhedra, general nonlinear reaction kineticsand general surface curvature, the method only relies on nodal values without needing additional evaluations usually associated with the quadrature of general reactionkinetics. This latter is particularly costly in 3D. The BS-VEM as implemented in thisstudy retains optimal convergence of second order in space.
Keywords: Bulk-surface PDEs; bulk-surface reaction-diffusion systems; polyhedral meshes; bulk-surface virtual element method; convergence.