Nonlinear Elliptic Boundary Value Problems Research Articles

Multi-grid continuation and spurious solutions for nonlinear eigenvalue problems

Abstract : Recently the author together with R. Bank has developed, implemented and successfully applied a continuation technique for the numerical solution of parameter-dependent nonlinear elliptic boundary value problems. The method was integrated into an existing multi-grid package based on an adaptive finite element discretization. This document present the continuation method and proves an important theoretical result for the corrector iteration. For the Bratu problem - delta u = lambda e sub u on the square with homogeneous Dirichlet conditions it is shown how spurious solutions may be encountered while computing relevant solutions, how the program handles those and how it allows to detect them.

Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems

In this note we present numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems in planar domains. There has recently been much progress in the development of such algorithms for linear elliptic problems ([1],[2],[3],[4]). These have focused on variational characterizations of the problem and on the preconditioning of the Schur complement associated with the decomposition. Although these could be used as part of a global Newton-type iterative scheme to solve a nonlinear problem, we choose the alternate path of first decomposing the problem and then applying an iterative method. Our motivation for this is two-fold; first, we expect it will lead to algorithms which will require less communication between subproblems, an attractive property for implementations on parallel processors; second, this approach has even been found to be more efficient for serial computations in some cases [6]. The essential step in this method is the solution of what we call the basic equations, a nonlinear analogue of the Schur complement problem. We are particularly concerned with the choice of boundary conditions at boundaries where subdomains intersect and their effect on the basic equations. Note that the methods we propose may also be applied to linear problems and seem to be novel in that context. We restrict ourselves here to the study of a decomposition into two domains:

Folds and cusps in Banach spaces with applications to nonlinear partial differential equations. II

Earlier the authors have given abstract properties characterizing the fold and cusp maps on Banach spaces, and these results are applied here to the study of specific nonlinear elliptic boundary value problems. Functional analysis methods are used, specifically, weak solutions in Sobolev spaces. One problem studied is the inhomogeneous nonlinear Dirichlet problem \[ Δ u + λ u − u 3 = g on Ω , u | ∂ Ω = 0 , \Delta u + \lambda u - {u^3} = g\quad {\text {on}}\;\Omega ,\qquad u|\partial \Omega = 0, \] where Ω ⊂ R n ( n ⩽ 4 ) \Omega \subset {{\mathbf {R}}^n}(n \leqslant 4) is a bounded domain. Another is a nonlinear elliptic system, the von Kármán equations for the buckling of a thin planar elastic plate when compressive forces are applied to its edge.

Game of two cars with containment probability as cost function

Abstract A stochastic version of the ‘game of two cars’ in which the cost function is a containment probability (Isaacs 1965) is dealt with. By numerically solving a nonlinear elliptic boundary value problem on a generalized torus in ℝ4, it is shown that a proportional navigation guidance law is an optimal feedback pursuit strategy.

Application of iteration functions to elliptic partial differential equations

Iteration functions discussed previously by Traub are first generalized to abstract spaces and this formulation is employed to obtain high-order solutions of linear and nonlinear elliptic boundary-value problems. For the application of the procedure, discretizations of two different orders are required. The numerical solutions of the higher-order discretized system are obtained by an iterative procedure which uses the solution of the lower-order system.

Compactness method in the finite element theory of nonlinear elliptic problems

Finite element approximation of a nonlinear elliptic pseudomonotone second-order boundary value problem in a bounded nonpolygonal domain Ω with mixed Dirichlet-Neumann boundary conditions is studied. In the discretization we approximate the domain Ω by a polygonal one, use linear conforming triangular elements and evaluate integrals by numerical quadratures. We prove the solvability of the discrete problem and on the basis of compactness properties of the corresponding operator (which is not monotone in general) we prove the convergence of approximate solutions to an exact weak solutionu∈H1 Ω). No additional assumption on the regularity of the exact solution is needed.

The Exact number of Solutions of Finite Difference Approximations of Some Nonlinear Elliptic Boundary Value Problems

We consider finite dimensional nonlinear eigenvalue problems of the type $Az = \lambda F(z)$ where A is a matrix,$(F(z))_j = f(z_j ),j = 1, \cdots ,n$. Such systems arise as discretizations of a corresponding boundary value problem. The number of solutions of both equations at a fixed $\lambda \in \mathbb{R}_ + $ may differ significantly, i.e., spurious solutions occur. We give an upper bound on the number of solutions at a fixed $\lambda $ in case of the nonlinearities $F(z) = \exp (z),f(z) = \sin z$, for polynomial and related nonlinearities. To this end we extend the equation into $C^n $ and use the Brouwer degree for complex analytic mappings. A homotopy argument yields that, roughly speaking, the number of solutions of the finite system is limited by the number of the solutions of the uncoupled system $z = \lambda F(z)$.

Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains

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.

Asymptotic Boundary Conditions and Numerical Methods for Nonlinear Elliptic Problems on Unbounded Domains

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.

Uniqueness and stability of strongly nonlinear elliptic boundary value problems

Uniqueness and stability of strongly nonlinear elliptic boundary value problems

Defect correction for nonlinear elliptic difference equations

The present paper is concerned with the study of a high-order defect correction technique for discretizations of nonlinear elliptic boundary value problems. The convergence of the method is analyzed in general and, in more detail, for a specific example. The algorithmic combination of defect correction and multigrid techniques is also discussed.

Multiplicity results for asymptotically homogeneous semilinear boundary value problems

This paper treats nonlinear elliptic boundary value problems of the form $$\Delta u + f(x,u) = 0in\Omega ,u = 0on\partial \Omega $$ in the spaceL2(Ω) by degree theoretic methods. Emphasis is placed on existence of multiplesolutions in the case, where the nonlinearity f crosses several eigenvalues of the correspondingeigenvalue problem Δθ+λθ = 0 with zero boundary values. No differentiability conditions(but Lipschitz type conditions) on f are assumed. A main tool is a new a priori bound forsolutions (Theorem 1). The method is not confined to the selfadjoint case. It applies alsoto some time-periodic parabolic and hyperbolic problems.

Regularity of solutions of nonlinear elliptic boundary value problems.

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

Finite element solution of nonlinear elliptic problems

The study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed Dirichlet-Neumann boundary conditions is presented. In the discretization variational crimes are commited (approximation of the given domain by a polygonal one, numerical integration). With the assumption that the corresponding operator is strongly monotone and Lipschitz-continuous and that the exact solutionu∈H1(Ω), the convergence of the method is proved; under the additional assumptionu∈H2(Ω), the rate of convergenceO(h) is derived without the use of Green's theorem.

Error estimates for mixed finite element approximations of nonlinear problems

The finite element methods have proved a very effective tool for the numerical solutions of nonlinear problems arising in elasticity and other related engineering sciences. Relative to linear elliptic theory, little is known about the accuracy and convergence properties of mixed finite element approximation of nonlinear elliptic boundary value problems. The nonlinear problems are much more complicated, since each problem has to be treated individually. This is one of the reasons that there is no unified and general theory for the nonlinear problems. In this paper, the application of the mixed finite element method to a highly nonlinear Dirichlet problem, which arises in the field of oceanography and elasticity is studied and new results involving the error estimates are derived. In fact, some of the results and methods to be described in this paper may be extended to more complicated problems or problems with other boundary conditions. As a special case, we obtain the well known error estimates for the corresponding linear and mildly nonlinear elliptic boundary value problems.

Critical behaviour of nonlinear elliptic boundary value problems suggested by exothermic reactions

SynopsisWe consider a class of semilinear elliptic boundary value problems depending on a parameter, which arise in the theory of combustion. Based on the results in another paper by the same author, a rigorous quantitative connection is shown between the solution set of the boundary value problem and that of a simple scalar equation (the Semenov approximation).

Numerical solution of non-linear elliptic boundary-value problems by isomorphic iterative methods

New implicit iterative methods are presented for the efficient numerical solution of non-linear elliptic boundary-value problems. Isomorphic iterative schemes in conjunction with preconditioning techniques are used for solving non-linear elliptic equations in two and three-space dimensions. The application of the derived methods on characteristic 2D and 3D non-linear boundary-value problems is discussed and numerical results are given.

S-Shaped bifurcation curves of nonlinear elliptic boundary value problems

S-Shaped bifurcation curves of nonlinear elliptic boundary value problems

Bunting identity and Mazur identity for non-linear elliptic systems including the black hole equilibrium problem

Recent work by G. Bunting and by P. O. Mazur has developed new techniques for proving uniqueness theorems for extensive classes of non-linear elliptic boundary value problems including that of the equilibrium state of an electromagnetically charged black hole. These methods are described and compared. It is shown that the rather general class of harmonic mappings that can be dealt with by the Bunting method (which needs no internal symmetry group) can be regarded as a generalisation of the particular (totally symetric) class of non-linearσ-models that can be dealt with by the Mazur method.

When are solutions to nonlinear elliptic boundary value problems convex?

We extend and apply a concavity maximum principle from [10, 9, 7] to some nonlinear elliptic boundary problems and free boundary problems on domains Ω⊂IRn. In particular we extend convex dead core' results from n = 2 as in [4 ] to arbitrary n. We also show the convexity of the coincidence set in the obstacle problem under suitable assumptions.