Second‐order partial differential equations (PDEs) in nondivergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton–Jacobi–Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second‐order PDEs. The nondivergence form in these problems is natural. If the coefficients of the diffusion matrix are not differentiable, the problem cannot be transformed into the more convenient variational form. We investigate tailored nonconforming finite element approximations of second‐order PDEs in nondivergence form, utilizing finite‐element Hessian recovery strategies to approximate second derivatives in the equation. We study both approximations with continuous and discontinuous trial functions. Of particular interest are a priori and a posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature.

中文翻译：

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二阶偏微分方程非变分形式的误差估计

考虑非散度形式的二阶偏微分方程（PDE）。在随机最优控制的情况下，这类方程通常作为Hamilton-Jacobi-Bellman方程解的子问题出现，或作为完全非线性的二阶PDE线性化而出现。这些问题中的不分歧形式是自然的。如果扩散矩阵的系数不可微，则无法将问题转化为更方便的变分形式。我们利用有限元Hessian恢复策略近似近似方程中的二阶导数，研究了非散度形式的二阶PDE的量身定制的非协调有限元逼近。我们用连续和不连续的试验函数研究两种近似值。特别令人感兴趣的是先验和后验误差估计以及自适应有限元方法。在数值实验中，我们的方法与文献中已知的其他方法进行了比较。