The Study Of Fast Algorithms For Nonlinear Elliptic Problems

Nonlinear elliptic partial differential equations are widely appeared in physics,mechanics and other fields,they usually have a large nu-merical of computational effort due to the nonlinearity.Therefore,the study of their efficient and fast algorithms has very important theo-retical significance and broad application prospects.This paper aims to investigate the fast computation of nonlinear elliptic problems by using extrapolation cascadic multigrid methods(EXCMG).First,EXCMG is combined with Newton method(abbreviated as EXCMG-Newton)to solve directly the semilinear and quasilinear elliptic equations.The numerical results show that this method can guarantee the standard finite element order of convergence.Secondly,EXCMG is combined with the multilevel linearization method(abbreviated as EXCMG-MLM)to solve a class of semilinear elliptic equations.For the case of the nonlinear term in the equation with bounded second derivative,the standard H1 and L2 finite element convergence and quasi computational optimality have been proved.The numerical results verify the theoretical results.In this paper,the theoretical and numerical results show that the multilevel linearization method can reduce the computational cost of nonlinear elliptic equations,and EXCMG has a better computational efficiency than the classical cascadic multigrid method due to the for-mer provided a better iterative initial value reducing the number of iterations.