On the Numerical Solution of Degenerate Linear and Nonlinear Elliptic Boundary Value Problems

We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis.

Consideration of an abstract improvement algorithm leads to the following principle, which is similar to that underlying iterative refinement: By making judicious use of relatively few high accuracy computations, high accuracy solutions can be obtained very efficiently by the algorithm. This principle is applied specifically to ${\text{GMRES}}(m)$ here; it can be similarly applied to a number of other “restarted” iterative linear methods as well. Results are given for numerical experiments in solving a discretized linear elliptic boundary value problem and in computing a step of an inexact Newton method using finite differences for a discretized nonlinear elliptic boundary value problem.

A survey on various computational techniques for nonlinear elliptic boundary value problems

Linear systems of equations with positive and symmetric matrices often occur in the numerical treatment of linear and nonlinear elliptic boundary value problems. If the CG algorithm is used to solve these equations, one is able to speed up the convergence by "preconditioning." The method of preconditioning with hierarchical basis has already been considered for the Laplace equation in two space dimensions and for linear conforming elements. In the present work this method is generalized to a large class of conforming and nonconforming elements.

Via the linking theorem, the existence of nontrivial solutions for a nonlinear elliptic Dirichlet boundary value problem with an inverse square potential is proved.

Trust-region methods for nonlinear elliptic equations with radial basis functions

The quasilinearization method is developed for strong solutions of semilinear and nonlinear elliptic boundary-value problems. We obtain two monotone, Lp-convergent sequences of approximate solutions. The order of convergence is two. The tools are some results on the abstract quasilinearization method and from weakly–near operators theory.

We consider symmetry properties of solutions to nonlinear elliptic boundary value problems defined on bounded symmetric domains of \({\mathbb R^n}\) . The solutions take values in ordered Banach spaces E, e.g. \({E=\mathbb R^N}\) ordered by a suitable cone. The nonlinearity is supposed to be quasimonotone increasing. By considering cones that are different from the standard cone of componentwise nonnegative elements we can prove symmetry of solutions to nonlinear elliptic systems which are not covered by previous results. We use the method of moving planes suitably adapted to cover the case of solutions of nonlinear elliptic problems with values in ordered Banach spaces.

In this paper, we deal with the application of Half-Sweep Accelerated Over Relaxation (HSAOR) method with nonlocal discretization scheme for solving two-dimensional nonlinear elliptic boundary value problems. To do this, we propose a new nonlocal arithmetic mean scheme namely the four-point rotated nonlocal arithmetic mean scheme being imposed into any nonlinear term in the proposed problems. By using the second order finite difference scheme, the half-sweep nonlinear approximation equation has been derived. Then, the nonlocal discretization scheme is applied to transform the system of nonlinear approximation equations into the corresponding system of linear equations. Throughout numerical results, it can be pointed out that the proposed HSAOR method was superior in terms of number of iterations, execution time and maximum error compared to Full-Sweep Successive Over-relaxation (FSSOR) and Half-Sweep Successive Over Relaxation (HSSOR).

Optimal summability of solutions to nonlinear elliptic problems

We present a derivation and implementation of asymptotic boundary conditions to be imposed on "artificial" boundaries for nonlinear elliptic boundary value problems on semi-infinite "cylindrical" domains. A general theory developed by the authors in [11] is applied to establish the existence of exact boundary conditions and then to obtain useful approximations to them. The derivation is based on the Laplace transform solution of the linearized problem at infinity. We discuss the incorporation of the asymptotic boundary conditions into a finite-difference scheme and present the results of numerical experiments on the solution of the Bratu problem in a two-dimensional stepped channel. We also touch on certain problems concerning the existence of solutions of this problem on infinite domains and conjecture on the behavior of the critical parameter value with respect to changes in the domain. Some numerical evidence supporting the conjecture is given.

On utilise des estimations de type Schauder pour ameliorer la regularite supposee des solutions de problemes aux limites non lineaires elliptiques

Turbulence is thought to play a key role in the transport of particles and energy within thermonuclear plasmas, and a number of codes have been developed to study the phenomena involved. A novel algorithm for calculating the force balance within such a code is presented. This involves the solution of a non-linear elliptic boundary value problem by considering it as a steady-state limit of a parabolic (heat) equation.

Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Bratu’s equation, Troesch’s problems) occurs engineering and science, including the modeling of chemical reactions diffusion processes and heat transfer. An analytical expression pertaining to the concentration of substrate is obtained using Homotopy perturbation method for all values of parameters. These approximate analytical results were found to be in good agreement with the simulation results.

The approximation of solutions of nonlinear elliptic boundary value problems having several solutions