Xu Jinchao mainly studies the numerical solution of partial differential equation, especially finite element, multi-mesh, and regional split method. 1. One of the most outstanding works of BPX pretreatment is the so-called BRAMBLE-PASCIAK-XU. PreconDi- Tioner, this work is in his doctoral paper, it is an existing multi-grid method One of the main components and is widely used by scientific computing communities, especially in parallel calculation. Xu Jinchao This pioneering work triggered a large number of successive research. 2. The main contributions of multiple meshes and regional splits are now in multiple mesh and regional split methods, and many basic theories are associated with Xu Jinchao. He earliest work in this regard (published in 1988 with BRAMBLE) is a well-known public issue in multiple grid theory. He proved uniform convergence of multi-grid method for the method of making a fixed step size for the method of non-symmetrical and non-elliptical boundary value. At the same time, he has established a basic framework for processing unstructured multi-grid issues, which later used from multisects to analyze a large number of multiple meshes and regional splits. In 1991, he worked with BRAMBLE and PASCIAK Junping, a uniform convergence theory of multi-sub-domain multi-Multiple Schwarz method, and created new technologies for analyzing multi-grid methods without elliptical formation. This work has led to the establishment of multi-grid theory of local encryption and interruption factors. Xu Jinchao later further developed the above theory, and established a unified framework for analyzing a large type of iterative method, including multiple mesh, regional split, and classic Jacobi? Gauss-SEIDEL iteration. These in-depth studies form the general principles of "spatial decomposition and sub-spatial correction". Published by this question is now the basic document in this area. Recently, Xu Jin super is committed to establishing unified theories of non-overlapping regional splitting methods. He has developed a universal technology that makes people have a more profound understanding of many different algorithms and to do more concise theoretical analysis. 3. Two-layer grid method for asymmetrical, non-linear issues In 1934, Xu Jinchao developed a general two-layer mesh method, so that the solution to the symmetrical positive problem is applied to non-true, non-linear problems, and The amount of operation is the same as the symmetrical positive problem. Recently, he combines his two grid ideas with nonlinear Galerkin methods, which is used to solve nonlinear parabolic problems. 4. New solutions and accessory subspace techniques of unable structs are obviously important to effectively apply multiple grid methods to general practical problems. To this end, Xu Jinchao has developed a new technique, which has been obtained for the best multi-grid pretreatment method for general non-structural grids. Its basic thinking is to use non-nest two-layer grid. The "thick" grid is structured, which can use existing methods, and "fine" mesh is only smooth. The framework of "Auxiliary Space Method" provides a general method for processing complex issues. The pretreatment of complex problems such as high-techings can be constructed and analyzed by corresponding simple issues (such as low elements). He also cooperated with Bank to develop a set of grid coarsering technologies with structural grid and analyzed multi-level basic matrix of unable structure grid.
