This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overline{\Omega}) \cap W^{2,n}_\mathrm{loc}(\Omega)$ to uniformly elliptic linear PDE of second order in nondivergence form with continuous leading coefficient in nonsmooth domains by finite element methods. These solutions satisfy the Alexandrov-Bakelman-Pucci (ABP) maximum principle, which provides an a~posteriori error control for $C^1$ conforming approximations. By minimizing this residual, we obtain an approximation to the solution $u$ in the $L^\infty$ norm. Although discontinuous functions do not satisfy the ABP maximum principle, this approach extends to nonconforming FEM as well thanks to well-established enrichment operators. Convergence of the proposed FEM is established for uniform mesh-refinements. The built-in a~posteriori error control (even for inexact solve) can be utilized in adaptive computations for the approximation of singular solutions, which performs superiorly in the numerical benchmarks in comparison to the uniform mesh-refining algorithm.

翻译：本文提出一种新方法，通过有限元方法逼近非散度形式一致椭圆二阶线性偏微分方程的强解 $u \in C(\overline{\Omega}) \cap W^{2,n}_\mathrm{loc}(\Omega)$，其中领先系数连续且定义域非光滑。此类解满足Alexandrov-Bakelman-Pucci（ABP）最大值原理，为 $C^1$ 协调逼近提供后验误差控制。通过最小化该残差，我们在 $L^\infty$ 范数下获得解 $u$ 的近似。尽管非连续函数不满足ABP最大值原理，但借助成熟的富化算子，该方法可拓展至非协调有限元。在一致网格细化下，所提有限元法的收敛性得以建立。内建的后验误差控制（即使对于非精确求解）可用于自适应计算中逼近奇异解，且数值基准测试表明，其性能优于一致网格细化算法。