[论文解读] On Approximating Discontinuous Solutions of PDEs by Adaptive Finite Elements

Shun Zhang|arXiv (Cornell University)|Jul 8, 2019

Differential Equations and Numerical Methods参考文献 12被引用 5

一句话总结

本文研究了使用自适应有限元方法求解偏微分方程的不连续解，表明在自适应网格上采用不连续分段常数逼近可消除过冲现象，而连续或不匹配的线性逼近则不可避免地产生非平凡的过冲。关键贡献在于证明了结合自适应网格加密与类似伽辽金的不连续逼近方法，对于尖锐层和不连续性问题是最优的。

ABSTRACT

For singularly perturbed problems with a small diffusion, when the transient layer is very sharp and the computational mesh is relatively coarse, the solution can be viewed as discontinuous. For both linear and nonlinear hyperbolic partial differential equations, the solution can be discontinuous. When finite element methods with piecewise polynomials are used to approximate these discontinuous solutions, numerical solutions often overshoot near a discontinuity. Can this be resolved by adaptive mesh refinements? In this paper, for a simple discontinuous function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is the best choice to achieve accuracy without overshooting. For discontinuous piecewise linear approximations, non-trivial overshootings will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a far away assumption, and non-trivial overshootings will always be observed under regular meshes. We calculate the explicit overshooting values for several typical cases. Several numerical tests are preformed for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper.

研究动机与目标

• 解决在具有尖锐层的PDE有限元逼近中，不连续性附近出现的数值过冲问题。
• 研究自适应网格加密是否能解决不连续解中的过冲问题。
• 比较在匹配与非匹配网格上，连续与不连续分段多项式逼近方法的性能。
• 量化标准有限元方法在双曲型与反应-扩散方程中过冲行为的表现。
• 建立利用自适应有限元方法对不连续解进行最优逼近的策略。

提出的方法

• 显式计算在不连续性匹配与非匹配网格上，对不连续函数的分段常数与线性投影。
• 分析在均匀与自适应网格下，连续与不连续有限元逼近的行为。
• 推导典型情况下过冲值的显式表达式，包括线性与常数逼近。
• 使用自适应网格加密使单元与不连续性对齐，从而最小化逼近误差与过冲。
• 对奇异摄动的反应-扩散方程与线性双曲方程进行数值测试，以验证理论结果。
• 在不同网格对齐条件下，对比连续、不连续以及线性与常数逼近方法的结果。

实验结果

研究问题

• RQ1 自适应网格加密能否消除PDE不连续解有限元逼近中的过冲？
• RQ2 网格与不连续性对齐程度如何影响分段常数与线性有限元逼近的精度与过冲？
• RQ3 为何在均匀网格上，连续分段线性逼近始终产生非平凡的过冲？
• RQ4 标准有限元逼近在典型不连续情况下的显式过冲值是什么？
• RQ5 在捕捉尖锐层方面，不连续分段常数逼近是否优于连续或线性逼近？

主要发现

• 在不连续性匹配网格上，不连续分段常数逼近不产生过冲，且对不连续解是最优的。
• 在非匹配网格上，结合自适应加密的不连续分段常数逼近可实现高精度且无过冲。
• 除非网格与不连续性完全对齐，否则不连续分段线性逼近仍表现出非平凡的过冲。
• 由于伽辽金公式中远端假设的存在，连续分段线性逼近在均匀网格上始终产生非平凡的过冲。
• 针对关键情况推导出显式过冲值，量化了标准有限元方法中的误差大小。
• 数值测试结果证实，采用不连续常数逼近的自适应加密方法能有效抑制奇异摄动与双曲型PDE中的过冲。