We present an efficient implementation of the double adaptivity algorithm of Cohen et al. (2012) within the setting of the Petrov–Galerkin method with optimal test functions. We apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard Galerkin finite element method. As an example, we demonstrate the feasibility of the method in the context of the convection-dominated diffusion problem. The presented ideas, however, apply to virtually any well-posed system of first-order partial differential equations, including singular perturbation problems.

中文翻译：

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双重适应范式

我们提出了Cohen等人的双适应算法的有效实现。（2012）在Petrov–Galerkin方法的设置中具有最佳测试功能。我们将此方法应用于用标准Galerkin有限元方法离散化的一般线性变分问题的超弱变分公式。例如，我们证明了该方法在对流为主扩散问题中的可行性。但是，所提出的思想实际上适用于任何一阶偏微分方程的系统，包括奇异摄动问题。