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：许多重要的偏微分方程的解满足界约束，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的约束条件。Chang和Nakshatrala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实现了这些界约束。本文为该方法的理论合理性提供了证明，涵盖高阶离散格式。我们证明了线性变分不等式的一个抽象最佳逼近结果，以及表明保界多项式具有与标准空间相当的逼近能力的估计。对于任意一个函数的无约束逼近，存在一个在W^{1,p}范数下可与其比较的约束逼近。在实际应用中，无法高效地表示和处理整个保界多项式族，但将界约束施加于Bernstein基多项式系数可确保该多项式满足这些约束。尽管我们的理论结果无法保证这类保界多项式子集具有高精度，但数值结果表明，在满足界约束的前提下，对于光滑解可达到最优精度阶数，且能清晰分辨对流扩散问题中的特征。