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Battling Gibbs phenomenon: On finite element approximations of discontinuous solutions of PDEs
Computers & Mathematics with Applications ( IF 2.9 ) Pub Date : 2022-08-02 , DOI: 10.1016/j.camwa.2022.07.014
Shun Zhang

In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view.

For a simple step 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 a good choice to achieve accuracy without overshoots. For discontinuous piecewise linear approximations, non-trivial overshoots 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 overshoots will always be observed under regular meshes. We calculate the explicit overshoot values for several typical cases. Numerical tests are conducted for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper. Also, we discuss the L1-minimization-based methods and do not recommend such methods due to their similar behavior to L2-based methods and more complicated implementations.



中文翻译:

对抗吉布斯现象:关于 PDE 不连续解的有限元近似

在本文中,我们想从逼近的角度阐明使用连续和不连续有限元来逼近不连续或几乎不连续 PDE 解时的吉布斯现象。

对于一个简单的阶跃函数,我们明确计算其在不连续匹配或非匹配网格上的连续和不连续分段常数或线性投影。对于简单的不连续对齐网格情况,分段不连续近似总是好的。对于一般不匹配的情况,我们解释了分段不连续常数逼近与自适应网格细化相结合是实现精度而没有超调的好选择。对于不连续的分段线性近似,将观察到非平凡的超调,除非网格与不连续性匹配。对于连续分段线性近似,计算基于“遥远的假设”,在规则网格下总是会观察到非平凡的超调。我们计算了几个典型案例的显式超调值。对奇异摄动反应扩散方程和线性双曲方程进行了数值测试,以验证我们在论文中的发现。另外,我们讨论大号1-基于最小化的方法,并且不推荐此类方法,因为它们的行为类似于大号2-基于方法和更复杂的实现。

更新日期:2022-08-02
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