Several Rectangular Finite Element Decomposition Schemes And Fast Algorithms For Biharmonic Equations

The biharmonic equations are widely used in many disciplines and application fields such as solid mechanics,fluid mechanics,and materials science.The finite element method is a commonly used numerical method for solving such equations.However,directly using the finite element method to solve biharmonic equations,there will be discrete al-gebra The system condition number difference(O(h^-4))and other defects.Therefore,it is of great theoretical significance and practical application value to design an efficient numerical solution algorithm for biharmonic problems.In this paper,the numerical solution of the biharmonic problem with homogeneous dirichlet boundary conditions on a rectangular grid.First,the idea of[1]decomposes the solution of the biharmonic equation on a rectangular grid into Poisson,Stokes and Poisson three second-order subproblems are solved,and four rectangular finite element solution methods are designed for these subproblems,and the condition number of the corresponding discrete system is improved to(O(h^-2)).Next,taking the fourth rectangu-lar finite element solution method as an example,the convergence of these finite element solution methods is proved through theoretical analysis.Finally,based on the algebraic multigrid method,it is the child of the above decomposition The corresponding fast solu-tion algorithm is designed for the problem,and numerical experiments verify the conver-gence and efficiency of the decomposition method.