非自邻接和不定椭圆方程 CDG 有限元的二阶超收敛性

IF 1.7 3区数学 Q2 MATHEMATICS, APPLIED

Advances in Computational Mathematics Pub Date : 2023-12-22 DOI:10.1007/s10444-023-10100-9

Xiu Ye, Shangyou Zhang

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引用次数: 0

摘要

摘要为求解二阶非自邻接和不定椭圆方程，设计了一种符合非连续伽勒金（CDG）有限元方法。与其他非连续 Galerkin（DG）方法不同的是，两个元素之间的边缘/三角形上的数值迹不是两个不连续 \(P_k\)函数的平均值，而是由四个（三维为八个）附近的 \(P_k\)函数提升的 \(P_{k+2}\)函数。所有现有的 DG 方法都有最佳收敛阶数，而这种 CDG 方法在求解一般二阶椭圆方程时，具有比最佳阶数高两阶的超收敛性。由于超收敛性，后处理可以将每个元素上的\(P_k\) CDG 解提升到准最优的\(P_{k+2}\) 解。二维和三维数值测试证实了这一理论。

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Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations

A conforming discontinuous Galerkin (CDG) finite element method is designed for solving second order non-self adjoint and indefinite elliptic equations. Unlike other discontinuous Galerkin (DG) methods, the numerical trace on the edge/triangle between two elements is not the average of two discontinuous \(P_k\) functions, but a lifted \(P_{k+2}\) function from four (eight in 3D) nearby \(P_k\) functions. While all existing DG methods have the optimal order of convergence, this CDG method has a superconvergence of order two above the optimal order when solving general second order elliptic equations. Due to the superconvergence, a post-process lifts a \(P_k\) CDG solution to a quasi-optimal \(P_{k+2}\) solution on each element. Numerical tests in 2D and 3D are provided confirming the theory.

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来源期刊

Advances in Computational Mathematics 数学-应用数学

CiteScore

3.00

自引率

5.90%

发文量

审稿时长

3 months

期刊介绍： Advances in Computational Mathematics publishes high quality, accessible and original articles at the forefront of computational and applied mathematics, with a clear potential for impact across the sciences. The journal emphasizes three core areas: approximation theory and computational geometry; numerical analysis, modelling and simulation; imaging, signal processing and data analysis. This journal welcomes papers that are accessible to a broad audience in the mathematical sciences and that show either an advance in computational methodology or a novel scientific application area, or both. Methods papers should rely on rigorous analysis and/or convincing numerical studies.