A new Galerkin finite element for the biharmonic equation is constructed on 2D rectangular and 3D cuboid meshes. In this <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</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">$k\geqslant 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>) interpolated Galerkin finite element construction, all unknowns associated with the interior of each element are determined by the direct interpolation of the right-hand-side function, and the rest of the unknowns, associated with the boundary of each element, are determined by solving the remaining linear equations of the Galerkin projection. In comparison with the traditional finite element method in two dimensions, our method reduces the number of unknowns from <InlineEquation ID="IEq4"> <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). Additionally, the method reduces the condition number drastically as it requires only 1% of the computer time, compared with the standard finite elements, in several numerical tests. We prove the existence and uniqueness of the solution and the optimal order of convergence. We confirm the theory by numerical tests in two dimensions and three dimensions.

Interpolated Galerkin Finite Elements on Rectangular and Cuboid Meshes for the Biharmonic Equation

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发表日期：

2025年1月28日

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Mengjiao Pan,
Tatyana Sorokina,
Shangyou Zhang

摘要

A new Galerkin finite element for the biharmonic equation is constructed on 2D rectangular and 3D cuboid meshes. In this $C^1$ $C^{1}$ - $Q_k$ $Q_{k}$ ( $k\geqslant 4$ $k ⩾ 4$ ) interpolated Galerkin finite element construction, all unknowns associated with the interior of each element are determined by the direct interpolation of the right-hand-side function, and the rest of the unknowns, associated with the boundary of each element, are determined by solving the remaining linear equations of the Galerkin projection. In comparison with the traditional finite element method in two dimensions, our method reduces the number of unknowns from $O(k^2)$ $O (k^{2})$ to O(k). Additionally, the method reduces the condition number drastically as it requires only 1% of the computer time, compared with the standard finite elements, in several numerical tests. We prove the existence and uniqueness of the solution and the optimal order of convergence. We confirm the theory by numerical tests in two dimensions and three dimensions.