凝聚粗网格上拟线性椭圆型偏微分方程的双网格$hp$型间断Galerkin有限元法
Two-grid $hp$-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes
摘要：
本文考虑扩展的两个网格$hp$版本的间断伽辽金有限元法的数值逼近的二阶拟线性椭圆边值问题的情况下，凝聚多边形/多面体网格的粗网格近似。我们记得，在两个网格设置，而它是必要的，以解决一个非线性问题的粗略近似空间，只有一个线性问题必须计算在原来的罚款有限元空间。在本文中，粗空间将通过聚集原始细网格的元素来构造。在这里，我们扩展了现有的先验和后验误差分析的两个网格$hp$版本的不连续伽辽金有限元法从10.1007/s10915-012-9644-1的粗网格组成的标准元形状，包括任意凝聚的粗网格。此外，我们开发了一个$hp$自适应两个网格算法，自适应设计的精细和粗糙的有限元空间，我们强调，这是在一个全自动的方式进行，因此可以被视为黑箱求解器。数值实验的二维和三维问题，以证明所提出的$hp$自适应两个网格方法的计算性能。
英文摘要：
This article considers the extension of two-grid $hp$-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglomerating elements from the original fine mesh. Here, we extend the existing a priori and a posteriori error analysis for the two-grid $hp$-version discontinuous Galerkin finite element method from 10.1007/s10915-012-9644-1 for coarse meshes consisting of standard element shapes to include arbitrarily agglomerated coarse grids. Moreover, we develop an $hp$-adaptive two-grid algorithm to adaptively design the fine and coarse finite element spaces; we stress that this is undertaken in a fully automatic manner, and hence can be viewed as blackbox solver. Numerical experiments are presented for two- and three-dimensional problems to demonstrate the computational performance of the proposed $hp$-adaptive two-grid method.
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