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.

翻译：本文提出了一种非线性有限元方法，其节点值能保持已知精确解的界。离散问题涉及一个非线性投影算子，将任意节点值映射为保界值，并在此投影的像空间中寻求数值解。由于该投影非单射，为恢复问题的适定性，需添加基于互补投影的稳定化项。在椭圆问题框架下，该离散问题可视为障碍问题的重构，通过Lipschitz投影引入不等式约束。以线性和非线性反应扩散问题为例，阐述了所提方法的推导过程。建立了合适范数下的近最优逼近结果。特别地，在线性情形下证明了数值解是所有节点保界有限元函数中能量范数意义下的最佳逼近。针对此类问题的一系列数值实验展示了所提保界有限元方法的优良性能。