Numerical methods for coupled systems of quasilinear elliptic equations with nonlinear boundary conditions

This paper is concerned with numerical solutions of a coupled system of arbitrary number of quasilinear elliptic equations under combined Dirichlet and nonlinear boundary conditions. A finite difference system for a transformed system of the quasilinear equations is formulated, and three monotone iterative schemes for the computation of numerical solutions are given using the method of upper and lower solutions. It is shown that each of the three monotone iterations converges to a minimal solution or a maximal solution depending on whether the initial iteration is a lower solution or an upper solution. A comparison result among the three iterative schemes is given. Also shown is the convergence of the minimal and maximal discrete solutions to the corresponding minimal and maximal solutions of the continuous system as the mesh size tends to zero. These results are applied to a heat transfer problem with temperature dependent thermal conductivity and a Lotka-Volterra cooperation system with degenerate diffusion. This degenerate property leads to some interesting distinct property of the system when compared with the non-degenerate semilinear systems. Numerical results are given to the above problems, and in each problem an explicit continuous solution is constructed and is used to compare with the computed solution     

Positive evanescent solutions of nonlinear elliptic equations

In this note we investigate the existence of positive solutions vanishing at +∞ to the elliptic equation Δu+f(x,u)+g(|x|)x⋅∇u=0, |x|>A>0, in Rn (n?3) under mild restrictions on the functions f, g.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

NontriviaL solutions of some nonlinear elliptic problems

This paper is concerned with a class of semilinear elliptic Dirichlet problems approximating degenerate equations. The aim is to prove the existence of at least 4^k?1 nontrivial solutions when the degeneration set consists of k distinct connected components     

Superlinear convergence theorems for Newton-type methods for nonlinear systems of equations

In this paper, someQ-order convergence theorems are given for the problem of solving nonlinear systems of equations when using very general finitely terminating methods for the solution of the associated linear systems. The theorems differ from those of Dembo, Eisenstat, and Steihaug in the different stopping condition and in their applicability to the nonlinear ABS algorithm.Lecture presented at the University of Bergamo, Bergamo, Italy, October 1989.     

Regularity for solutions of nonlinear elliptic equations with degenerate coercivity

We prove regularity results for solutions of some nonlinear Dirichlet problems for an equation in the form

Nonexistence of positive solutions to nonlinear nonlocal elliptic systems

In this paper we consider the question of nonexistence of nontrivial solutions for nonlinear elliptic systems involving fractional diffusion operators. Using a weak formulation approach and relying on a suitable choice of test functions, we derive sufficient conditions in terms of space dimension and systems parameters. Also, we present three main results associated to three different classes of systems.     

Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives

In this paper, by introducing the fractional derivative in the sense of Caputo, the Adomian decomposition method is directly extended to study the coupled Burgers equations with time- and space-fractional derivatives. As a result, the realistic numerical solutions are obtained in a form of rapidly convergent series with easily computable components. The figures show the effectiveness and good accuracy of the proposed method.     

Multiplicity of solutions for resonant elliptic systems

We establish the existence and multiplicity of solutions for some resonant elliptic systems. The results are proved by applying minimax arguments and Morse theory.     

Gradient estimates for viscosity solutions of singular fully nonlinear elliptic equations

We establish Lipschitz regularity for solutions to a family of non-isotropic fully nonlinear partial differential equations of elliptic type. In general such a regularity is optimal. No sign constraint is imposed on the solution, thus limiting free boundaries may have two-phases. Our estimates are then employed in combination with fine regularizing techniques to prove existence of viscosity solutions to singular nonlinear PDEs.     

Three positive solutions for nonlinear elliptic equations in finite strip with hole

In this paper, we study the effect of the domain on the multiplicity of positive solutions for the nonlinear elliptic equation. Using the Ekeland variational principle and the center mass function, we prove that there are at least three positive solutions for the nonlinear elliptic equation in finite strip with holes.     

Steklov-Neumann eigenproblems and nonlinear elliptic equations with nonlinear boundary conditions

We study the solvability of nonlinear second order elliptic partial differential equations with nonlinear boundary conditions. We introduce the notion of “eigenvalue-lines” in the plane; these eigenvalue-lines join each Steklov eigenvalue to the first eigenvalue of the Neumann problem with homogeneous boundary condition. We prove existence results when the nonlinearities involved asymptotically stay, in some sense, below the first eigenvalue-lines or in a quadrilateral region (depicted in Fig. 1) enclosed by two consecutive eigenvalue-lines. As a special case we derive the so-called nonresonance results below the first Steklov eigenvalue as well as between two consecutive Steklov eigenvalues. The case in which the eigenvalue-lines join each Neumann eigenvalue to the first Steklov eigenvalue is also considered. Our method of proof is variational and relies mainly on minimax methods in critical point theory.     

Parallel iterative methods using factorized preconditioning matrices for solving elliptic equations on triangular grids

Parallel analogs of the variants of the incomplete Cholesky-conjugate gradient method and the modified incomplete Cholesky-conjugate gradient method for solving elliptic equations on uniform triangular and unstructured triangular grids on parallel computer systems with the MIMD architecture are considered. The construction of parallel methods is based on the use of various variants of ordering the grid points depending on the decomposition of the computation domain. Results of the theoretic and experimental studies of the convergence rate of these methods are presented. The solution of model problems on a moderate number processors is used to examine the efficiency of the proposed parallel methods.     

Robust Schur complement method for strongly anisotropic elliptic equations

We present robust and asymptotically optimal iterative methods for solving 2D anisotropic elliptic equations with strongly jumping coefficients, where the direction of anisotropy may change sharply between adjacent subdomains. The idea of a stable preconditioning for the Schur complement matrix is based on the use of an exotic non‐conformal coarse mesh space and on a special clustering of the edge space components according to the anisotropy behavior. Our method extends the well known BPS interface preconditioner [2] to the case of anisotropic equations. The technique proposed also provides robust solvers for isotropic equations in the presence of degenerate geometries, in particular, in domains composed of thin substructures. Numerical experiments confirm efficiency and robustness of the algorithms for the complicated problems with strongly varying diffusion and anisotropy coefficients as well as for the isotropic diffusion equations in the ‘brick and mortar’ structures involving subdomains with high aspect ratios. Copyright © 1999 John Wiley & Sons, Ltd.     

Adaptive frame methods for elliptic operator equations

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are especially interested in discretization schemes based on frames. The central objective is to derive an adaptive frame algorithm which is guaranteed to converge for a wide range of cases. As a core ingredient we use the concept of Gelfand frames which induces equivalences between smoothness norms and weighted sequence norms of frame coefficients. It turns out that this Gelfand characteristic of frames is closely related to their localization properties. We also give constructive examples of Gelfand wavelet frames on bounded domains. Finally, an application to the efficient adaptive computation of canonical dual frames is presented.     

Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.