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 $O((k+1)^2)$, 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 $L^2$ theory. Furthermore, all FVE schemes lose local conservation properties over boundary dual elements when dealing with Dirichlet boundary conditions. The proposed FVE-2L 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 $L^2$ 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 $H^1$ and $L^2$ 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.

翻译：本文提出了一种双层对偶策略，用于构建一类新型高阶有限体积元（FVE-2L）格式，该格式能够避免现有高阶有限体积元（FVE）格式的主要常见缺陷。现有高阶FVE格式的构造过程较为复杂，因为其构造中每个主单元内使用的对偶单元数量以$O((k+1)^2)$的速率增长，其中$k$为格式的阶数。此外，所有$k$阶FVE格式对$L^2$理论所需的正则性要求（$H^{k+2}$）高于其逼近理论的要求。而且，所有FVE格式在处理Dirichlet边界条件时，在边界对偶单元上会丧失局部守恒性质。所提出的FVE-2L格式构造更为简单，因为每个主单元内仅包含固定数量（四个）的对偶单元。该格式还将$L^2$理论的正则性要求降低至$H^{k+1}$，并在第二对偶层的所有对偶单元上保持了通量形式和方程形式的局部守恒律。本文证明了该格式的稳定性以及$H^1$和$L^2$收敛性。数值结果展示了FVE-2L格式的收敛性与守恒特性。此外，对于Laplacian算子，FVE-2L格式刚度矩阵的条件数增长速率与现有FVE格式及有限元格式具有相同量级。