LOCALLY CONSERVATIVE SERENDIPITY FINITE ELEMENT SOLUTIONS FOR ELLIPTIC EQUATIONS

In this paper, we post-process an eight-nodes-serendipity finite element solution for elliptic equations. In the post-processing procedure, we first construct a control volume for each node in the serendipity finite element mesh, then we enlarge the serendipity finite element space by adding some appropriate element-wise bubbles and require the novel solution to satisfy the local conservation law on each control volume. Our post-processing procedure can be implemented in a parallel computing environment and its computational cost is proportional to the cardinality of the serendipity elements. Moreover, both our theoretical analysis and numerical examples show that the postprocessed solution converges to the exact solution with optimal convergence rates both under H-1 and L-2 norms. A numerical experiment for a single-phase porous media problem validates the necessity of the post-processing procedure.