The <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$Q_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) interpolated Galerkin finite elements are constructed on 2D and 3D rectangular meshes for the Poisson equation. In this interpolated Galerkin finite element method, all nodal-value degrees of freedom of the standard <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$Q_k$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>k</mi> </msub> </math></EquationSource> </InlineEquation> finite element inside a rectangle/cube are replaced by the Laplacian values at the nodes. Thus one obtains these degrees of freedom of the finite element solution by the interpolation values of the right hand side function. The remaining system of linear equations consists of unknowns at inter-element boundaries only. That is, the interpolated finite element solution is the Galerkin projection in a much smaller subspace. The method reduces the number of unknowns of the traditional finite element method from <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$O(k^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to O(k) per element in 2D. We prove the unique solution converges at the optimal order in both <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> norms. Numerical examples are computed by the interpolated finite elements and the traditional finite elements in 2D and 3D.

Interpolated Galerkin finite elements on rectangular and cuboid meshes for the Poisson equation

期刊论文
发表日期：

2025年2月17日

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Xiuli Wang,
Ran Zhang,
Shangyou Zhang

摘要

The $C^0$ $C^{0}$ - $Q_k$ $Q_{k}$ ( $k\ge 2$ $k \geq 2$ ) interpolated Galerkin finite elements are constructed on 2D and 3D rectangular meshes for the Poisson equation. In this interpolated Galerkin finite element method, all nodal-value degrees of freedom of the standard $Q_k$ $Q_{k}$ finite element inside a rectangle/cube are replaced by the Laplacian values at the nodes. Thus one obtains these degrees of freedom of the finite element solution by the interpolation values of the right hand side function. The remaining system of linear equations consists of unknowns at inter-element boundaries only. That is, the interpolated finite element solution is the Galerkin projection in a much smaller subspace. The method reduces the number of unknowns of the traditional finite element method from $O(k^2)$ $O (k^{2})$ to O(k) per element in 2D. We prove the unique solution converges at the optimal order in both $L^2$ $L^{2}$ and $H^1$ $H^{1}$ norms. Numerical examples are computed by the interpolated finite elements and the traditional finite elements in 2D and 3D.