Structure-preserving Nonlinear Filtering for Continuous and Discontinuous Galerkin Spectral/hp Element Methods
Structure-preserving Nonlinear Filtering for Continuous and Discontinuous Galerkin Spectral/hp Element Methods
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DOI：
10.1137/20m1337223
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发表时间：
2021-01
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影响因子：
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通讯作者：
Vidhi Zala;R. Kirby;A. Narayan
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文献类型：
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作者：
Vidhi Zala;R. Kirby;A. Narayan
有限元模拟已被用于求解各种模拟物理、化学和生物现象的偏微分方程（PDEs）。得到的微分方程离散解往往不满足必要的物理性质，如正性或单调性。这种无效的解决方案既带来了建模挑战，因为模拟结果的物理解释是不可能的，也带来了计算挑战，因为可能需要这些属性来推进方案。因此，我们考虑保留这些结构解决方案属性的计算解决方案的问题，我们将其作为解决方案的附加约束强制执行。我们特别考虑了一类凸约束，它包括正性和单调性。通过嵌入这些约束作为后处理凸优化过程，我们能够计算满足一般类型凸约束的解决方案。对于某些类型的约束（包括正性和单调性），优化是一个过滤器，即一个减少范数的操作。我们对一维时间相关偏微分方程进行了各种测试，证明了该方法的有效性，并且我们通过经验表明，收敛速度不受约束的影响。
Finite element simulations have been used to solve a variety of partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural solution properties, which we enforce as additional constraints on the solution. We consider in particular the class of convex constraints, which includes positivity and monotonicity. By embedding such constraints as a postprocessing convex optimization procedure, we are able to compute solutions that satisfy general types of convex constraints. For certain types of constraints (including positivity and monotonicity), the optimization is a filter, i.e., a norm-decreasing operation. We provide a variety of tests on one-dimensional time-dependent PDEs that demonstrate the efficacy of the method, and we empirically show that rates of convergence are unaffected by the inclusion of the constraints.