At the fully discrete setting, stability of the discontinuous Petrov--Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators. We construct such operators for $H^1$ and $\boldsymbol{H}(\mathrm{div})$ on simplices in any space dimension and arbitrary polynomial degree. The resulting test spaces are smaller than previously analyzed cases. For parameter-dependent norms, we achieve uniform boundedness by the inclusion of exponential layers. As an example, we consider a canonical DPG setting for reaction-dominated diffusion. Our test spaces guarantee uniform stability and quasi-optimal convergence of the scheme. We present numerical experiments that illustrate the loss of stability and error control by the residual for small diffusion coefficient when using standard polynomial test spaces, whereas we observe uniform stability and error control with our construction.
翻译:在完全离散的环境下,具有最佳测试功能的不连续性Petrov-Galerkin(DPG)方法的稳定需要当地测试空间,以确保Fortin操作员的存在。我们为任何空间尺寸和任意多球度的简化分子建造了1美元和1美元(mathrm{div})的操作员。由此形成的测试空间小于以前分析的个案。对于依赖参数的规范,我们通过纳入指数层来实现统一的约束。举例来说,我们考虑为反应主导扩散设置一个卡通式的DPG设置。我们的测试空间保证了统一稳定性和准最佳组合。我们提出了数字实验,说明在使用标准多球试验空间时,小扩散系数的剩余部分会失去稳定性和误差控制,而我们则在构建时看到统一的稳定性和误控。