New Finite Volume Element Schemes Based on a Two-Layer Dual Strategy (CSIAM Transactions on Applied Mathematics, 2025)
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. Theexisting high-order FVE schemes are complicated to construct since the number ofthe dual elements in each primary element used in their construction increases witha rate $\mathcal{O}((k+1)^2),$ where $k$ is the order of the scheme. Moreover, all $k$-th-order FVEschemes require a higher regularity $H^{k+2}$ than the approximation theory for the $L^2$ theory. Furthermore, all FVE schemes lose local conservation properties over boundarydual elements when dealing with Dirichlet boundary conditions. The proposed FVE-2L schemes has a much simpler construction since they have a fixed number (four) ofdual elements in each primary element. They also reduce the regularity requirementfor the $L^2$ theory to $H^{k+1}$ and preserve the local conservation law on all dual elementsof the second dual layer for both flux and equation forms. Their stability and $H^1$ and $L^2$ convergence are proved. Numerical results are presented to illustrate the convergenceand conservation properties of the FVE-2L schemes. Moreover, the condition numberof the stiffness matrix of the FVE-2L schemes for the Laplacian operator is shown tohave the same growth rate as those for the existing FVE and finite element schemes.