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：在全离散设定下，使用最优测试函数的间断Petrov-Galerkin（DPG）方法的稳定性要求局部测试空间确保Fortin算子的存在性。我们针对任意空间维度和任意多项式阶数的单纯形上的$H^1$和$\boldsymbol{H}(\mathrm{div})$空间构造了此类算子。所得测试空间比此前分析的情况更小。对于参数依赖范数，我们通过引入指数层实现了均匀有界性。作为示例，我们考虑了反应主导扩散问题的典型DPG设定。我们的测试空间保证了该方案的均匀稳定性和拟最优收敛性。我们展示了数值实验，揭示了当扩散系数较小时使用标准多项式测试空间会出现稳定性丧失和残差误差控制失效的情况，而我们的构造则能实现均匀稳定性和误差控制。