Finite element approximation for uniformly elliptic linear PDE of second order in nondivergence form
This work provides a novel error control technique for finite element approximations of nondivergence form PDEs, which is important for computational mathematics but represents an incremental advance over existing FEM approaches.
The paper introduces a finite element method for approximating strong solutions to uniformly elliptic linear PDEs in nondivergence form, using the ABP maximum principle for a posteriori error control. The method achieves superior performance in adaptive computations for singular solutions compared to uniform mesh refinement.
This paper proposes a novel technique for the approximation of strong solutions $u \in C(\overlineΩ) \cap W^{2,n}_\mathrm{loc}(Ω)$ 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.