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Newton's iteration with a conjugate gradient based decomposition method for an elliptic PDE with a nonlinear boundary condition

Jonas Koko (2004)

International Journal of Applied Mathematics and Computer Science

Newton's iteration is studied for the numerical solution of an elliptic PDE with nonlinear boundary conditions. At each iteration of Newton's method, a conjugate gradient based decomposition method is applied to the matrix of the linearized system. The decomposition is such that all the remaining linear systems have the same constant matrix. Numerical results confirm the savings with respect to the computational cost, compared with the classical Newton method with factorization at each step.

Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations

Monnanda Erappa Shobha, Ioannis K. Argyros, Santhosh George (2014)

Applicationes Mathematicae

We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is y δ with | | y - y δ | | ≤ δ , K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where F ' ( x ₀ ) - 1 exists (F’(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator. The parameter...

Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations

Qin Li, Qun Lin, Hehu Xie (2013)

Applications of Mathematics

The paper deals with error estimates and lower bound approximations of the Steklov eigenvalue problems on convex or concave domains by nonconforming finite element methods. We consider four types of nonconforming finite elements: Crouzeix-Raviart, Q 1 rot , E Q 1 rot and enriched Crouzeix-Raviart. We first derive error estimates for the nonconforming finite element approximations of the Steklov eigenvalue problem and then give the analysis of lower bound approximations. Some numerical results are presented to...

Nonconforming Galerkin methods based on quadrilateral elements for second order elliptic problems

Jim Douglas Jr., Juan E. Santos, Dongwoo Sheen, Xiu Ye (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Low-order nonconforming Galerkin methods will be analyzed for second-order elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P1, as for conforming elements; however, it is necessary to introduce new elements in the rectangular case. Optimal order error estimates are demonstrated in all cases with respect to a broken norm in H1(Ω)...

Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

Peter Oswald (2017)

Applications of Mathematics

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H 1 norm best approximation error estimates for H 2 functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate...

Nonhomogeneous boundary conditions and curved triangular finite elements

Alexander Ženíšek (1981)

Aplikace matematiky

Approximation of nonhomogeneous boundary conditions of Dirichlet and Neumann types is suggested in solving boundary value problems of elliptic equations by the finite element method. Curved triangular elements are considered. In the first part of the paper the convergence of the finite element method is analyzed in the case of nonhomogeneous Dirichlet problem for elliptic equations of order 2 m + 2 , in the second part of the paper in the case of nonhomogeneous mixed boundary value problem for second order...

Nonlinear boundary value problems with application to semiconductor device equations

Miroslav Pospíšek (1994)

Applications of Mathematics

The paper deals with boundary value problems for systems of nonlinear elliptic equations in a relatively general form. Theorems based on monotone operator theory and concerning the existence of weak solutions of such a system, as well as the convergence of discretized problem solutions are presented. As an example, the approach is applied to the stationary Van Roosbroeck’s system, arising in semiconductor device modelling. A convergent algorithm suitable for solving sets of algebraic equations generated...

Nonobtuse tetrahedral partitions that refine locally towards Fichera-like corners

Larisa Beilina, Sergey Korotov, Michal Křížek (2005)

Applications of Mathematics

Linear tetrahedral finite elements whose dihedral angles are all nonobtuse guarantee the validity of the discrete maximum principle for a wide class of second order elliptic and parabolic problems. In this paper we present an algorithm which generates nonobtuse face-to-face tetrahedral partitions that refine locally towards a given Fichera-like corner of a particular polyhedral domain.

Currently displaying 1261 – 1280 of 2193