论文标题
适用于毕业生方程式的自适应不连续的彼得 - 盖尔金方法
An adaptive discontinuous Petrov-Galerkin method for the Grad-Shafranov equation
论文作者
论文摘要
在这项工作中，我们建议并为非线性毕业生 - shafranov方程提出并开发一种任意阶段的自适应不连续的彼得 - 加盖尔（DPG）方法。根据最小残留方法给出了该方程的DPG方案的超视公式。与常规的有限元方法相比，DPG方案的优点是提供更准确的梯度，这对于Grad-Shafranov方程的数值解决方案是需要的。使用自适应网格细化方法增强了数值方案，并开发了基于剩余方法中的剩余标准，以实现动态优化。探索了生成系统的非线性求解器，并发现使用安德森加速度的PICARD迭代可以有效地解决该系统。最后，提出的算法使用域分解方法在MFEM上并行实施，我们的实现是一般的，支持了任意的准确性和一般网格。提出了数值结果以证明所提出算法的效率和准确性。
In this work, we propose and develop an arbitrary-order adaptive discontinuous Petrov-Galerkin (DPG) method for the nonlinear Grad-Shafranov equation. An ultraweak formulation of the DPG scheme for the equation is given based on a minimal residual method. The DPG scheme has the advantage of providing more accurate gradients compared to conventional finite element methods, which is desired for numerical solutions to the Grad-Shafranov equation. The numerical scheme is augmented with an adaptive mesh refinement approach, and a criterion based on the residual norm in the minimal residual method is developed to achieve dynamic refinement. Nonlinear solvers for the resulting system are explored and a Picard iteration with Anderson acceleration is found to be efficient to solve the system. Finally, the proposed algorithm is implemented in parallel on MFEM using a domain-decomposition approach, and our implementation is general, supporting arbitrary order of accuracy and general meshes. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithm.
