  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.
