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

研究者和他的同事们研究了超收敛现象及其在科学和工程中的几个主要计算方法的恢复。本文的研究目标包括：（1）发展间断Galerkin方法的超收敛恢复技术和相应的误差估计器;（2）利用恢复技术提高特征值逼近;（3）研究p型谱配置方法的超收敛现象。本课题将采用有限元超收敛、间断Galerkin方法、偏微分方程理论中的区域变分技术和逼近理论中的经典结果等最新的数学理论，对计算算法的自适应设计具有重要意义，并可直接应用于工程计算。该项目的成功将弥合工程实践和数学理论发展之间的差距，并扩大科学界的知识。该项目具有坚实的多学科和跨学科的内容和广泛的应用在软件行业。

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.