Iterative methods for $k$-Hessian equations

Local solvability of the k-Hessian equations

In this work, we study the existence of local solutions in R to k-Hessian equation, for which the nonhomogeneous term f is permitted to change the sign or be non negative; if f is C∞ , so is the local solution. We also give a classification for the second order polynomial solutions to the k−Hessian equation, it is the basis to construct the local solutions and obtain the uniform ellipticity of ...

Iterative Methods for Linear and Nonlinear Equations

Iterative Methods for Nonlinear Elliptic Equations

‖f(u)− f(v)‖−1 ≤ L‖u− v‖−1. We drop the dependence of f on x to emphasis the nonlinearity in u. Let Vh ⊂ H 0 (Ω) be the linear finite element space associated to a quasi-uniform mesh Th. The nonlinear Garlerkin method is: find uh ∈ Vh such that (2) Auh = f(uh) in Vh, where A : H 0 (Ω)→ H−1(Ω) is the elliptic operator defined by −∆. We shall assume the existence and locally uniqueness of the sol...

ABS-Type Methods for Solving $m$ Linear Equations in $frac{m}{k}$ Steps for $k=1,2,cdots,m$

‎The ABS methods‎, ‎introduced by Abaffy‎, ‎Broyden and Spedicato‎, ‎are‎‎direct iteration methods for solving a linear system where the‎‎$i$-th iteration satisfies the first $i$ equations‎, ‎therefore a‎ ‎system of $m$ equations is solved in at most $m$ steps‎. ‎In this‎‎paper‎, ‎we introduce a class of ABS-type methods for solving a full row‎‎rank linear equations‎, ‎w...

On Preconditioned Iterative Methods for Burgers Equations

We study the Newton method and a fixed-point method for solving the system of nonlinear equations arising from the Sinc-Galerkin discretization of the Burgers equations. In each step of the Newton method or the fixedpoint method, a structured subsystem of linear equations is obtained and needs to be solved numerically. In this paper, preconditioning techniques are applied to solve such linear s...