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：本文提出一种新颖技术，通过有限元方法逼近非光滑区域上具有连续主系数的非散度形式一致椭圆二阶线性偏微分方程的强解$u \in C(\overline{\Omega}) \cap W^{2,n}_\mathrm{loc}(\Omega)$。这些解满足Alexandrov-Bakelman-Pucci（ABP）最大值原理，该原理为$C^1$协调逼近提供了后验误差控制。通过最小化该残差，我们获得在$L^\infty$范数下对解$u$的逼近。尽管非连续函数不满足ABP最大值原理，但借助成熟的富集算子，该方法同样可推广至非协调有限元。对于一致网格细化，所提出的有限元方法的收敛性得以确立。其内置的后验误差控制（即使在不精确求解情况下）可用于自适应计算中以逼近奇异解，在数值基准测试中，该方法相比一致网格细化算法表现出更优性能。