TY - JOUR
T1 - New Finite Volume Element Schemes Based on a Two-Layer Dual Strategy
AU - Huang , Weizhang
AU - Wang , Xiang
AU - Zhang , Xinyuan
JO - CSIAM Transactions on Applied Mathematics
VL - 3
SP - 527
EP - 554
PY - 2025
DA - 2025/09
SN - 6
DO - http://doi.org/10.4208/csiam-am.SO-2024-0051
UR - https://global-sci.org/intro/article_detail/csiam-am/24374.html
KW - Finite volume, two-layer dual mesh, conservation, $L^2$
estimate, minimum angle condition.
AB - <p style="text-align: justify;">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 $\mathcal{O}((k+1)^2),$ 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.</p>