For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.

翻译：在数值逼近中，将偏微分方程重新表述为残量最小化问题具有以下优势：所得线性系统对称正定，且残量范数可提供后验误差估计器。此外，该方法还能处理一般非齐次边界条件。然而，在许多最小残量公式中，残量的一个或多个项是在负或分数阶Sobolev范数下进行度量的。本文提出一种通用方法，在保证数值解拟最优性的前提下，将这些范数替换为可高效计算的表达式。我们通过验证具有非齐次Dirichlet和/或Neumann边界条件的二阶椭圆型模型方程的四种公式所需满足的inf-sup条件，展示了该方法的可行性。针对带有混合非齐次Dirichlet和Neumann边界条件的泊松问题，我们在超弱一阶系统公式下开展了数值实验并报告相关结果。