Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the $W^{1,p}$ norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints.

翻译：许多重要偏微分方程的解满足界约束条件，但通过有限元或有限差分方法计算得到的近似解通常无法满足相同条件。Chang与Nakshatrala提出通过求解变分不等式而非线性变分问题，在有限元方法中强制执行此类界约束。本文为该方法的理论合理性提供证明，包括高阶离散格式。我们证明了线性变分不等式的抽象最佳逼近定理，并给出了相关估计，表明满足界约束的多项式具有与标准空间相当的逼近能力。对于任意无约束函数逼近，存在一个在$W^{1,p}$范数下精度可比的有约束逼近。实际计算中，人们无法高效表示和操控全体满足界约束的多项式族，但将界约束施加于Bernstein基多项式系数可确保该多项式本身满足约束。尽管本文的理论结果无法保证此类子集具有高精度，数值实验表明：在界约束条件下，光滑解可获得最优收敛阶，而对流扩散问题的特征分辨率保持锐利。