New Finite Volume Element Schemes Based on a Two-Layer Dual Strategy

Abstract

A two-layer dual strategy is proposed in this work to construct a new family of high-order finite volume element (FVE-2L) schemes that can avoid main common drawbacks of the existing high-order finite volume element (FVE) schemes. The existing high-order FVE schemes are complicated to construct since the number of the dual elements in each primary element used in their construction increases with a rate $\mathcal{O}\left((k+1{)}^{2}\right)$, where k is the order of the scheme. Moreover, all k-th-order FVE schemes require a higher regularity H^k+2 than the approximation theory for the L² theory. Furthermore, all FVE schemes lose local conservation properties over boundary dual elements when dealing with Dirichlet boundary conditions. The proposed FVE2 L schemes has a much simpler construction since they have a fixed number (four) of dual elements in each primary element. They also reduce the regularity requirement for the L² theory to H^k+1 and preserve the local conservation law on all dual elements of the second dual layer for both flux and equation forms. Their stability and H¹ and L² convergence are proved. Numerical results are presented to illustrate the convergence and conservation properties of the FVE-2L schemes. Moreover, the condition number of the stiffness matrix of the FVE-2L schemes for the Laplacian operator is shown to have the same growth rate as those for the existing FVE and finite element schemes.

Keywords

Finite volume / two-layer dual mesh / conservation / L² estimate / minimum angle condition