A Finite Volume Element Solution Based on Postprocessing Technique Over Arbitrary Convex Polygonal Meshes

Abstract

. A special (cid:12)nite volume element method based on postprocessing technique is proposed to solve the anisotropic di(cid:11)usion problem on arbitrary convex polygonal meshes. The shape function of polygonal (cid:12)nite element method is constructed by Wachspress generalized barycentric coordinate, and by adding some element-wise bubble functions to the (cid:12)nite element solution, we get a new (cid:12)nite volume element solution that satis(cid:12)es the local conservation law on a certain dual mesh. The postprocessing algorithm only needs to solve a local linear algebraic system on each primary cell, so that it is easy to implement. More interesting is that, a general construction of the bubble functions is introduced on each polygonal cell, which enables us to prove the existence and uniqueness of the post-processed solution on arbitrary convex polygonal meshes with full anisotropic di(cid:11)usion tensor. The optimal H 1 and L 2 error estimates of the post-processed solution are also obtained. Finally, the local conservation property and convergence of the new polygonal (cid:12)nite volume element solution are veri(cid:12)ed by numerical experiments.