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.

中文翻译：

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一种节点守界有限元方法

这项工作提出了一种非线性有限元方法，其节点值保留精确解已知的边界。离散问题涉及非线性投影算子，将任意节点值映射到保界值，并在此投影范围内寻求数值解。由于投影不是单射的，因此添加基于互补投影的稳定性以恢复适定性。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 投影纳入不等式约束。该方法的推导针对线性和非线性反应扩散问题进行了举例说明。建立了合适规范中最接近的近似结果。特别是，我们证明，在线性情况下，数值解是所有节点守界有限元函数中能量范数的最佳近似。针对此类问题的一系列数值实验展示了所提出的保界有限元方法的良好行为。