In this work we give a complete error analysis of the Discontinuous Petrov Galerkin (DPG) method, accounting for all the approximations made in its practical implementation. Specifically, we consider the DPG method that uses a trial space consisting of polynomials of degree $p$ on each mesh element. Earlier works showed that there is a "trial-to-test" operator $T$, which when applied to the trial space, defines a test space that guarantees stability. In DPG formulations, this operator $T$ is local: it can be applied element-by-element. However, an infinite dimensional problem on each mesh element needed to be solved to apply $T$. In practical computations, $T$ is approximated using polynomials of some degree $r > p$ on each mesh element. We show that this approximation maintains optimal convergence rates, provided that $r\ge p+N$, where $N$ is the space dimension (two or more), for the Laplace equation. We also prove a similar result for the DPG method for linear elasticity. Remarks on the conditioning of the stiffness matrix in DPG methods are also included.
The Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Chair for Computational Analysis of Technical Systems, RWTH Aachen University（亚琛工业大学技术系统计算分析讲席）；Center for Computing Research, Sandia National Laboratories（桑迪亚国家实验室计算研究中心）
A locking free hp DPG method for linear elasticity with symmetric stresses
The Institute for Computational Engineering and Sciences, the University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；Department of Mathematics and Statistics, Portland State University（波特兰州立大学数学与统计系）；The Institute for Mathematics and its Applications, University of Minnesota（明尼苏达大学数学及其应用研究所）
本文在间断 Petrov-Galerkin (DPG) 框架下提出了两种新方法，用于线性弹性问题，同时获得应力和位移近似在网格尺寸 h 和多项式次数 p 上的最优精度，并研究了无锁收敛性质。
High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations
The Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin（德克萨斯大学奥斯汀分校计算工程与科学研究所）；MSC Software Corporation（MSC软件公司）
Smooth Bezier Surfaces over Arbitrary Quadrilateral Meshes
The Rachel and Selim Benin School of Computer Science and Engineering（Rachel和Selim Benin计算机科学与工程学）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Microsoft Corp.（微软公司）
