ANALYSIS OF OPTIMAL SUPERCONVERGENCE OF LOCAL DISCONTINUOUS GALERKIN METHOD FOR ONE-DIMENSIONAL LINEAR PARABOLIC EQUATIONS
ANALYSIS OF OPTIMAL SUPERCONVERGENCE OF LOCAL DISCONTINUOUS GALERKIN METHOD FOR ONE-DIMENSIONAL LINEAR PARABOLIC EQUATIONS
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发表时间：
2013
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通讯作者：
Yang Yang-Yang;Chi-Wang Shu
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作者：
Yang Yang-Yang;Chi-Wang Shu
本文研究一维线性抛物线方程的局部间断伽辽金（LDG）有限元法在使用交变通量时误差的超收敛性。我们证明，如果我们应用分段 k 次多项式，LDG 解与精确解之间的误差在具有适当初始离散化的 Radau 点处是 (k + 2) 阶超收敛。此外，我们还证明 LDG 解对于精确解的特定投影的误差是 (k + 2) 阶超收敛的。尽管我们只考虑周期性边界条件，但该边界条件并不是必需的，因为我们不使用傅立叶分析。我们的分析对于任意规则网格和任意 k ≥ 1 的 Pk 多项式都是有效的。我们进行数值实验来证明本文证明的超收敛率是最优的。
In this paper, we study the superconvergence of the error for the local discontinuous Galerkin (LDG) finite element method for one-dimensional linear parabolic equations when alternating flux is used. We prove that if we apply piecewise k-th degree polynomials, the error between the LDG solution and the exact solution is (k + 2)-th order superconvergent at the Radau points with suitable initial discretization. Moreover, we also prove the LDG solution is (k + 2)-th order superconvergent for the error to a particular projection of the exact solution. Even though we only consider periodic boundary condition, this boundary condition is not essential, since we do not use Fourier analysis. Our analysis is valid for arbitrary regular meshes and for Pk polynomials with arbitrary k ≥ 1. We perform numerical experiments to demonstrate that the superconvergence rates proved in this paper are optimal.