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 $O((k+1)^2)$, 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.

Figures (8)

• Figure 1: Dual elements/regions on the reference element for single-layer FVEs (cf. Wang.2016).
• Figure 2: Examples of dual elements (shaded regions, $K_{\mathrm{I}}^{*}$ (left) and $K_{\mathrm{II}}^{*}$ (right)) for each dual layer.
• Figure 3: Interpolation nodes on the reference element $\hat{K}$. The colored points refer to the interpolation nodes on $Q_1$ (red), $Q_2$ (green), $Q_3$ (blue), and $Q_4$ (yellow). Points with two colors are shared freedoms.
• Figure 4: The conservation properties of the FVE-2L schemes in Example \ref{['ex:Elliptic problem']}. The dual elements marked blue indicate that local conservation is maintained, while the dual elements marked yellow indicate the violation of the local conservation law. A mesh with $h \approx 1/8$ is used.
• Figure 5: The numerical convergence rates of FVE-2L schemes for Example \ref{['ex:Elliptic problem']}. Here, $N$ denotes the number of intervals in each axis direction of the primary mesh.

Theorems & Definitions (5)

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