| Abstract: |
| Conservation laws are fundamental physical properties that are expected to be preserved in numerical discretizations. We propose a two-layered dual strategy for the finite volume element method (FVEM), which possesses the conservation laws in both flux form and equation form. In particular, for problems with Dirichlet boundary conditions, the proposed schemes preserves conservation laws on all triangles, whereas conservation properties may be lost on boundary dual elements by existing vertex-centered finite volume element schemes.
Theoretically, we carry out the optimal $L^2$ analysis with reducing the regularity requirement from $u\in H^{k+2}$ to $u\in H^{k+1}$. While, as a comparison, all existing $L^2$ results for high-order $(k>=2)$ finite volume element schemes require $u\in H^{k+2}$ in the analysis. |
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