This paper is concerned with goal-oriented a posteriori error estimation for nonlinear functionals in the context of nonlinear variational problems solved with continuous Galerkin finite element discretizations. A two-level, or discrete, adjoint-based approach for error estimation is considered. The traditional method to derive an error estimate in this context requires linearizing both the nonlinear variational form and the nonlinear functional of interest which introduces linearization errors into the error estimate. In this paper, we investigate these linearization errors. In particular, we develop a novel discrete goal-oriented error estimate that accounts for traditionally neglected nonlinear terms at the expense of greater computational cost. We demonstrate how this error estimate can be used to drive mesh adaptivity. We show that accounting for linearization errors in the error estimate can improve its effectivity for several nonlinear model problems and quantities of interest. We also demonstrate that an adaptive strategy based on the newly proposed estimate can lead to more accurate approximations of the nonlinear functional with fewer degrees of freedom when compared to uniform refinement and traditional adjoint-based approaches.

翻译：本文关注于非线性变分问题中非线性泛函的目标导向后验误差估计，该类问题采用连续伽辽金有限元离散求解。我们考虑基于两级（即离散）伴随方法的误差估计。在此框架下，传统误差估计方法需要同时线性化非线性变分形式与感兴趣的非线性泛函，这会在误差估计中引入线性化误差。本文系统研究了这些线性化误差，特别地，我们发展了一种新型离散目标导向误差估计，该估计以更高计算成本为代价，考虑了传统上被忽略的非线性项。我们展示了该误差估计如何驱动网格自适应，并证明在误差估计中计入线性化误差能提升多个非线性模型问题及感兴趣量的估计有效性。数值实验表明，相较于均匀加密网格和传统伴随方法，基于新提出估计的自适应策略能以更少自由度获得非线性泛函的更精确近似。