This work introduces finite element methods for a class of elliptic fully nonlinear partial differential equations. They are based on a minimal residual principle that builds upon the Alexandrov--Bakelman--Pucci estimate. Under rather general structural assumptions on the operator, convergence of $C^1$ conforming and discontinuous Galerkin methods is proven in the $L^\infty$ norm. Numerical experiments on the performance of adaptive mesh refinement driven by local information of the residual in two and three space dimensions are provided.

翻译：本文针对一类完全非线性椭圆偏微分方程提出了有限元方法。这些方法基于一个最小残差原理，该原理建立在Alexandrov--Bakelman--Pucci估计之上。在算子满足相当一般的结构性假设下，本文证明了$C^1$协调有限元方法和间断Galerkin方法在$L^\infty$范数下的收敛性。同时，提供了在二维和三维空间中，基于残差局部信息驱动的自适应网格加密的性能数值实验。