We study a fourth-order div problem and its approximation by the discontinuous Petrov-Galerkin method with optimal test functions. We present two variants, based on first and second-order systems. In both cases we prove well-posedness of the formulation and quasi-optimal convergence of the approximation. Our analysis includes the fully-discrete schemes with approximated test functions, for general dimension and polynomial degree in the first-order case, and for two dimensions and lowest-order approximation in the second-order case. Numerical results illustrate the performance for quasi-uniform and adaptively refined meshes.

翻译：我们研究第四阶div问题及其与具有最佳测试功能的不连续的Petrov-Galerkin方法的近似值,我们根据第一和第二阶系统提出两种变式,在这两种情况下,我们证明近似值的配制和近似最佳趋同性。我们的分析包括具有近似测试功能的完全分解计划,在第一阶案件中涉及一般尺寸和多级,在第二阶案件中涉及两个维度和最低级近似值。数字结果说明了准统一和适应性改进的模贝的性能。