Finite element simulations have been used to solve various 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 can 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 method's efficacy, and we empirically show that rates of convergence are unaffected by the inclusion of the constraints.

中文翻译：

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连续和不连续伽辽金谱/hp 元素方法的结构保持非线性滤波

有限元模拟已被用于求解模拟物理、化学和生物现象的各种偏微分方程 (PDE)。由此产生的 PDE 离散解通常不满足必要的物理特性，例如正性或单调性。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，因为可能需要这些特性来推进方案。因此，我们考虑保留这些结构解属性的计算解的问题，我们将其作为对解的附加约束。我们特别考虑凸约束的类别，包括正性和单调性。通过嵌入诸如后处理凸优化程序之类的约束，我们可以计算满足一般类型凸约束的解。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，即范数减少操作。我们提供了对一维时间相关 PDE 的各种测试，证明了该方法的有效性，并且我们凭经验表明收敛速度不受包含约束的影响。