A nodally bound-preserving finite element method
A nodally bound-preserving finite element method
复制标题
一种节点守界有限元方法
DOI：
10.1093/imanum/drad055
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发表时间：
2023
影响因子：
2.1
通讯作者：
Barrenechea G
中科院分区：
文献类型：
--
作者：
Barrenechea G
本文提出了一种节点值保持精确解已知界的非线性有限元方法。离散问题涉及一个非线性投影算子，该算子将任意节点值映射为保界的节点值，并在该投影范围内寻求数值解。由于投影不是内射的，所以增加了基于互补投影的稳定化，以便恢复适定性。在椭圆问题的框架内，离散问题可以被视为离散障碍问题的重新表述，通过Lipschitz投影结合了不等式约束。文中以线性和非线性反应扩散问题为例说明了该方法的推导过程。建立了在适当范数下的近最佳逼近结果。特别地，我们证明了在线性情形下，数值解是所有节点型保界有限元函数中能量范数的最佳逼近。对这类问题的一系列数值实验表明，所提出的保界有限元方法具有良好的性能。
This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones and seeks the numerical solution in the range of this projection. As the projection is not injective, a stabilisation based upon the complementary projection is added in order to restore well-posedness. Within the framework of elliptic problems, the discrete problem may be viewed as a reformulation of a discrete obstacle problem, incorporating the inequality constraints through Lipschitz projections. The derivation of the proposed method is exemplified for linear and nonlinear reaction-diffusion problems. Near-best approximation results in suitable norms are established. In particular, we prove that, in the linear case, the numerical solution is the best approximation in the energy norm among all nodally bound-preserving finite element functions. A series of numerical experiments for such problems showcase the good behaviour of the proposed bound-preserving finite element method.