A mountain pass method for the numerical solution of semilinear wave equations

Summary It is well-known that periodic solutions of semilinear wave equations can be obtained as critical points of related functionals. In the situation that we studied, there is usually an obvious solution obtained as a solution of linear problem. We formulate a dual variational problem in such a way that the obvious solution is a local minimum. We then find additional non-obvious solutions via a numerical mountain pass algorithm, based on the theorems of Ambrosetti, Rabinowitz and Ekeland. Numerical results are presented.Research supported in part by grant DMS-9208636 from the National Science FoundationResearch supported in part by grant DMS-9102632 from the National Science Foundation     

A cascadic multigrid algorithm for semilinear elliptic problems

Summary. We propose a cascadic multigrid algorithm for a semilinear elliptic problem. The nonlinear equations arising from linear finite element discretizations are solved by Newton's method. Given an approximate solution on the coarsest grid on each finer grid we perform exactly one Newton step taking the approximate solution from the previous grid as initial guess. The Newton systems are solved iteratively by an appropriate smoothing method. We prove that the algorithm yields an approximate solution within the discretization error on the finest grid provided that the start approximation is sufficiently accurate and that the initial grid size is sufficiently small. Moreover, we show that the method has multigrid complexity. Received February 12, 1998 / Revised version received July 22, 1999 / Published online June 8, 2000     

Hybrid mountain pass homoclinic solutions of a class of semilinear elliptic PDEs

Variational gluing arguments are employed to construct new families of solutions for a class of semilinear elliptic PDEs. The main tools are the use of invariant regions for an associated heat flow and variational arguments. The latter provide a characterization of critical values of an associated functional. Among the novelties of the paper are the construction of “hybrid” solutions by gluing minima and mountain pass solutions and an analysis of the asymptotics of the gluing process.     

On computation of solution curves for semilinear elliptic problems

We consider computation of solution curves for semilinear elliptic equations. In case solution is stable, we present an algorithm with monotone convergence, which is a considerable improvement of the corresponding schemes in [4] and [5]. For the unstable solutions, we show how to construct a fourth-order evolution equation, for which the same solution will be stable.     

Positive mountain pass solutions for a semilinear elliptic equation with a sign-changing weight function

We use the mountain pass theorem to study the existence and multiplicity of positive solutions of the generalisation of the well-known logistic equation -Δu=λg(x)u(x)(1-u(x))$-\mathrm{\Delta }u=\lambda g(x)u(x)(1-u(x))$ with Dirichlet boundary conditions to the case where g changes sign.     

Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems

In this paper, we prove a mountain pass theorem in order intervals in which the position of the mountain pass point is given precisely in terms of the order structure. By using this result and constructing special flows, we deal with the existence of multiple solutions and sign-changing solutions for the following classes of elliptic Dirichlet boundary value problems: (1) nonlinear terms have concave property near zero and have superlinear but subcritical growth at infinity; (2) nonlinear terms are of the formh(x)f(u), withh(x) changing sign; (3) the asymptotically linear case. We obtain several new existence results of nodal solutions and give more comparable relations among the positive, negative and sign-changing solutions obtained. Our method is set up in an abstract setting and should be useful in other problems. Dedicated to P.H. Rabinowitz on the occasion of his 60th birthday This work was carried out while the first author was visiting Utah State University. He acknowledges the support from a NSF grant of the U.S.A. and from the Chinese National Science Foundation. Supported in part by a NSF grant.     

On accelerated monotone iterations for numerical solutions of semilinear elliptic boundary value problems

This paper is concerned with the computational algorithms for finite difference solutions of a class of semilinear elliptic boundary value problems. An accelerated monotone iterative scheme is presented by using the method of upper and lower solutions. The rate of convergence of the iterations is estimated by the infinity norm, and the rate of convergence is quadratic for a larger class of nonlinear functions, including monotone nonincreasing functions. An application is given to a logistic model problem in ecology.     

Wavelets collocation methods for the numerical solution of elliptic BV problems

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.     

Local mountain pass for a class of elliptic system

In this paper we are concerned with the existence and concentration of positive solutions for the following class of elliptic system

     

A numerical algorithm for the solution of Signorini problems

In this paper we propose a new iterative algorithm for the solution of a certain class of Signorini problems. Such problems arise in the modelling of a variety of physical phenomena and usually involve the determination of an unknown free boundary. Here we describe a way of locating the free boundary directly and provide a proof that the algorithm converges when used with analytic methods. The advantage of this algorithm is that it can be used in conjunction with any numerical method with minimal development of extra code. We demonstrate its application with the boundary element method to some physical problems in both two and three dimensions.     

Maximum principles for functionals associated with the solution of semilinear elliptic boundary value problems

We consider a new P-function associated with the solutionu of an elliptic boundary value problem and obtain pointwise bounds for the gradient in terms of the maximum ofu and the geometry of the domain. SimilarP-functions have previously been used to obtain bounds of the same type. Our results give improved bounds for certain problems, in particular we obtain isoperimetric inequalities for the maximum stress in the Saint-Venant torsion problem.     

Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems

We study the numerical approximation of distributed optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the L ²-error estimates are of order o(h), which is optimal according with the $C^{0,1}(\overline{\Omega})$ -regularity of the optimal control.     

Multiple solutions for inhomogeneous critical semilinear elliptic problems

In this paper, we consider the existence of multiple positive solutions for an inhomogeneous critical semilinear elliptic problem. We show that the problem possesses at least four positive solutions.     

Nonselfadjoint semilinear elliptic boundary value problems

Summary In this paper we study the existence of solutions of nonselfadjoint semilinear elliptic boundary value problems with a bounded nonlinear term. We emphasize that this nonlinear term may depend on the derivatives of the function in a nontrivial way. In the proof of our main result we use the Leray-Schauder degree theory.Supported in part by the C.A.I.C.Y.T., Ministry of Education (Spain), under Grant no. 3258/83.     

A mountain pass algorithm with projector

In this paper, we present an enhanced version of the minimax algorithm of Chen, Ni, and Zhou that offers the additional guarantee that the solution found is a fix-point of a projector on a cone. Positivity, negativity, and monotonicity can be expressed in this way. The convergence of the algorithm is proved by means of a “computational deformation lemma” instead of the usual deformation lemma used in the calculus of variations.