SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 718-748, April 2024.
Abstract. 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 [math] and [math] 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 face bubble functions that are polynomials on faces and decay exponentially in the interior. 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.

中文翻译：

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强大的 DPG 测试空间和 Fortin 算子——[数学]和[数学]案例

《SIAM 数值分析杂志》，第 62 卷，第 2 期，第 718-748 页，2024 年 4 月。
摘要。在完全离散设置下，具有最优测试函数的不连续 Petrov-Galerkin (DPG) 方法的稳定性需要局部测试空间来确保 Fortin 算子的存在。我们在任何空间维度和任意多项式次数的单纯形上构造 [math] 和 [math] 的此类运算符。由此产生的测试空间比之前分析的案例要小。对于参数相关范数，我们通过包含面气泡函数来实现一致有界性，这些面气泡函数是面上的多项式并在内部呈指数衰减。作为示例，我们考虑反应主导扩散的规范 DPG 设置。我们的测试空间保证了该方案的一致稳定性和准最优收敛。我们提出了数值实验，说明当使用标准多项式测试空间时，小扩散系数的残差会导致稳定性和误差控制的损失，而我们通过我们的构造观察到均匀的稳定性和误差控制。