Superconvergence of p-version/spectral collocation, discontinuous Galerkin methods and eigenvalue approximation

研究者和他的同事研究了科学和工程中几种主要计算方法的超收敛现象及其恢复。研究目标包括：(1)建立不连续Galerkin方法的超收敛恢复技术和相关误差估计；(2)利用恢复技术增强特征值近似；(3)研究p版/谱配置方法的超收敛现象。本项目将采用有限元超收敛中的一些最新数学理论、不连续伽辽金方法、偏微分方程理论中的域变分技术和逼近理论中的经典结果。该研究对计算算法的自适应设计具有重要意义，在工程计算中具有直接的应用价值。该项目的成功将弥合工程实践与数学理论发展之间的差距，并拓宽科学界的知识。本项目内容扎实，多学科交叉，在软件行业应用广泛。

The investigator and his colleagues study the superconvergence phenomenon and its recovery of several major computational methods in science and engineering. The research objectives include (1) to develop a superconvergence recovery technique and associated errorestimator for discontinuous Galerkin method; (2) to enhance the eigenvalue approximation by recovery techniques; and (3) to investigate superconvergence phenomena of p-version/spectral collocation methods. Some recent mathematical theory in finite element superconvergence, discontinuous Galerkin methods, as well as domain variation techniques in the PDE theory and classical results in approximation theory will be employed in the project.The study is of great importance for the adaptive design ofcomputational algorithms and has a direct application inengineering computation. The success of the project will bridge a gap between engineering practice and mathematical theoretical development and widen the knowledge in the scientific community. The project has solid multi- and interdisciplinary contents and wide application in the software industry.