Newton flows for elliptic functions I

Trigonometric functions, elliptic functions, elliptic modular forms

A Newton-Like Method for Convex Functions

A Newton-like method for convex functions is derived. It is shown that this method can be better than the Newton method. Especially good results can be obtained if we combine these two methods. Illustrative numerical examples are given. Mathematics Subject Classification: 65H05

Altered Jacobian Newton Iterative Method for Nonlinear Elliptic Problems

We present an Altered Jacobian Newton Iterative Method for solving nonlinear elliptic problems. Effectiveness of the proposed method is demonstrated through numerical experiments. Comparison of our method with Newton Iterative Method is also presented. Convergence of the Newton Iterative Method is highly sensitive to the initialization or initial guess. Reported numerical work shows the robustn...

Heun functions versus elliptic functions

We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard’s generalization of Floquet’s theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integr...

Global Inexact Newton Multilevel FEM for Nonlinear Elliptic Problems

The paper deals with the multilevel solution of elliptic partial differential equations (PDEs) in a finite element setting: uniform ellipticity of the PDE then goes with strict monotonicity of the derivative of a nonlinear convex functional. A Newton multigrid method is advocated, wherein linear residuals are evaluated within the multigrid method for the computation of the Newton corrections. T...