Adaptive, second-order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source
Adaptive, second-order in time, primitive-variable discontinuous Galerkin schemes for a Cahn–Hilliard equation with a mass source
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DOI：
10.1093/imanum/dru035
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发表时间：
2015-07
影响因子：
2.1
通讯作者：
A. Aristotelous;O. Karakashian;S. Wise
中科院分区：
文献类型：
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作者：
A. Aristotelous;O. Karakashian;S. Wise
两个完全离散的，不连续的Galerkin格式的时间动态，局部细化网格在空间中开发的四阶Cahn-Hilliard方程与一个添加的非线性反应项，一个现象学模型，可以描述癌症肿瘤的生长。建议的计划，这两个二阶的时间，是基于连续变量的不连续Galerkin空间公式，是有效的，在任何数量的空间维度。我们证明了该计划是收敛的，最优阶误差界，即使在网格随时间变化的情况下，网格变化的数量是由一些常数的限制。该计划灵活，高阶精度的替代标准的混合C 0有限元方法和板（板）有限元方法求解四阶抛物型偏微分方程。最后，我们的文件与测试显示该计划的收敛性在预测的速度和灵活性的方法，有效地近似复杂的解决方案的动态。
Two fully discrete, discontinuous Galerkin schemes with time-dynamic, locally refined meshes in space are developed for a fourth-order Cahn–Hilliard equation with an added nonlinear reaction term, a phenomenological model that can describe cancerous tumour growth. The proposed schemes, which are both second-order in time, are based on a primitive-variable discontinuous Galerkin spatial formulation that is valid in any number of space dimensions. We prove that the schemes are convergent, with optimal-order error bounds, even in the case where the mesh is changing with time, provided that the number of mesh changes is bounded by some constant. The schemes represent flexible, high-order accurate alternatives to the standard mixed C0finite element methods and nonconforming (plate) finite element methods for solving fourth-order parabolic partial differential equations. We conclude the paper with tests showing the convergence of the scheme at the predicted rates and the flexibility of the method for approximating complex solution dynamics efficiently.