Research On EXCMG Method Based On Interpolation Coefficient Finite Element

Nonlinear elliptic partial differential equations(pdes)are widely used in science,technology and engineering.Newton method and its variants are classical methods for solving nonlinear problems with second-order convergence.However,the convergence results depend heavily on the initial value.Therefore,good initial value and accelerated are very important for solving nonlinear problem.In this paper,we discuss how to solve semilinear elliptic problems by extrapolating cascadic multigrid method based on interpolation coefficient finite element method.First,this paper discretize the semilinear elliptic equation with the linear interpolation coefficient finite element method.The discrete equations can be written in the form of a numerical matrix multiplied by a vector function independent of the unknowns,which can be computed in one go,independent of the unknowns.In this way,the solution of the discrete equations can only focus on the iteration of the node value of the vector function,thus saving a lot of time.The convergence of the linear interpolation coefficient finite element in the sense of zero modulus is proved.Secondly,for the nonlinear algebraic equations discretized by the linear interpolation coefficient finite element,a new algorithm,extrapolated cascadic multigrid method(Newton-EXCMG),is presented.then the convergence of the extrapolated cascadic multigrid method based on interpolation coefficient is discussed.Finally,the extrapolated cascadic multigrid method based on the finite element interpolation coefficient is used to solve the semilinear elliptic problem with constant coefficient,the semilinear elliptic problem with variable coefficients,the poisson boltzmann equation and the sine Gordon equation.Numerical results show that the error between the initial value provided by EXCMG and the accurate finite element solution on each mesh layer reaches the 3rd order convergence in the sense of discrete L2 norm.Since EXCMG provides a good initial value of iteration,the solution of discrete equations only needs Newton iteration once(i.e.calculate the Newton tangent matrix once)which equivalent to solving a linear problem,the amount of calculation is greatly reduced.The results of numerical test proved the efficiency of the algorithm.