Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.     

A full multigrid method for semilinear elliptic equation

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.     

ON MULTIPLE POSITIVE SOLUTIONS FOR A NONLINEAR ELLIPTIC PROBLEM IN R^N

61. IntroductionConsider the semilinear elliptic problemwhere p E (1, ac) for N 2 3, and p E (l, co) for N = 1, 2, 0 S a(x) E C(R"), A > 0 is areal parameter.The existence and uniqueness of solution for such problems have been considered by mailyauthors recently (see [l--4] and references therein). In I3], T. Bartsch and Wang, Z. Q. provedthat with more genaral nolilineaxities (1.1) has at least one solution for A large under someconditions on a(x), one of which is that a--1 (0) has non…     

Finding multiple solutions to elliptic systems with polynomial nonlinearity

Elliptic systems with polynomial nonlinearity usually possess multiple solutions. In order to find multiple solutions, such elliptic systems are discretized by eigenfunction expansion method (EEM). Error analysis of the discretization is presented, which is different from the error analysis of EEM for scalar elliptic equations in three aspects: first, the choice of framework for the nonlinear operator and the corresponding isomorphism of the linearized operator; second, the definition of an auxiliary problem in deriving the relation between the L² norm and H¹ norm of the Ritz projection error; third, the bilinearity/nonbilinearity of the linearized variational forms. The symmetric homotopy for the discretized equations preserves not only D₄ symmetry, but also structural symmetry. With the symmetric homotopy, a filter strategy and a finite element Newton refinement, multiple solutions to a system of semilinear elliptic equations arising from Bose–Einstein condensate are found.     

Quasilinear elliptic eigenvalue problems

Summary The generalized Palais-Smale condition introduced in [26] is applied to obtain multiple solutions of variational eigen-value problems with quasilinear principal part, thereby extending some well-known existence results for semilinear elliptic problems. This research was supported by the Sonderforschungsbereich 72 of the Deutsche Forschungsgemeinschaft.     

A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints

In this paper optimal control problems governed by elliptic semilinear equations and subject to pointwise state constraints are considered. These problems are discretized using finite element methods and a posteriori error estimates are derived assessing the error with respect to the cost functional. These estimates are used to obtain quantitative information on the discretization error as well as for guiding an adaptive algorithm for local mesh refinement. Numerical examples illustrate the behavior of the method.     

Asymptotic behavior of positive solutions of semilinear elliptic equations in ℝ n , II

In this paper, we will analyze further to obtain a finer asymptotic behavior of positive solutions of semilinear elliptic equations in R^n by employing the Li＇s method of energy function.     

SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENTS FOR ELLIPTIC PROBLEMS ON TRIANGULATION

In this paper, we present the least-squares mixed finite element method and investigate superconvergence phenomena for the second order elliptic boundary-value problems over triangulations. On the basis of the L~2-projection and some mixed finite element projections, we obtain the superconvergence result of least-squares mixed finite     

Four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities

In this paper, we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation. With the help of the Nehari manifold and the center mass function, we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole.     

Multiplicity of positive solutions for semilinear elliptic problems in unbounded domains

In this paper, we study the effect of domain shape on the multiplicity of positive solutions for the semilinear elliptic equations. We prove a Palais-Smale condition in unbounded domains and assert that the semilinear elliptic equation in unbounded domains has multiple positive solutions.     

Sub-harmonicity,monotonicity formula and finite Morse index solutions of an elliptic equation with negative exponent

A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with "negative exponent" is established.It is well known that such a monotonicity formula plays an essential role in the study of finite Morse index solutions of equations with "positive exponent".Unlike the positive exponent case,we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.     

A finite element method for computing the bifurcation function for semilinear elliptic BVPs

The method of alternative problems can be used to show that a semilinear elliptic boundary value problem (Lu + g(x, u) = 0 with g_u(x, u) bounded below) is equivalent to a finite-dimensional problem ( , ), in the sense that their solution sets, which are not necessarily singletons, are in a one-to-one correspondence. This correspondence is based on a map σ from low-frequency to high-frequency Fourier components of solutions. A numerical method is presented for approximating σ and hence also solutions of the BVP. The method uses finite element approximations and avoids the use of eigenfunction expansions. Existence, uniqueness, and error estimates for the approximations of σ and solutions u are derived.     

Morse indices and exact multiplicity of solutions to semilinear elliptic problems

We obtain precise global bifurcation diagrams for both one-sign and sign-changing solutions of a semilinear elliptic equation, for the nonlinearity being asymptotically linear. Our method combines the bifurcation approach and spectral analysis.

     

Error estimates of the lumped mass finite element method for semilinear elliptic problems

We are concerned with the semilinear elliptic problems. We first investigate the L²-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory.     

A cascadic multigrid method for a kind of semilinear elliptic problem

In this paper, we analyze a cascadic multigrid method for semilinear elliptic problems in which the derivative of the semilinear term is Hölder continuous. We first investigate the standard finite element error estimates of this kind of problem. We then solve the corresponding discrete problems using the cascadic multigrid method. We prove that the algorithm has an optimal order of convergence in energy norm and quasi-optimal computational complexity. We also report some numerical results to support the theory.     

Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations

In this article, we investigate the application of pseudo‐transient‐continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, use the PTC‐methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction‐type PTC‐method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC ‐Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2005–2022, 2017     

Fucik Spectrum, Sign-Changing, and Multiple Solutions for Semilinear Elliptic Boundary Value Problems with Resonance at Infinity

In this paper, we use Fucik spectrum, ordinary differential equation theory of Banach spaces to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, especially with resonance at infinity, and obtain new results on the existence of multiple solutions and sign-changing solutions with a weakening of the P.S. condition. In one case we get up to seven nontrivial solutions. The techniques have independent interest.     

Exact multiplicity and global structure of solutions for a class of semilinear elliptic equations

In this paper, we establish an exact multiplicity result of solutions for a class of semilinear elliptic equation. We also obtain a precise global bifurcation diagram of the solution set. As a result, an open problem presented by C.-H. Hsu and Y.-W. Shih [C.-H. Hsu, Y.-W. Shih, Solutions of semilinear elliptic equations with asymptotic linear nonlinearity, Nonlinear Anal. 50 (2002) 275-283] is completely solved. Our argument is mainly based on bifurcation theory and continuation method.     

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.     

Error Estimates and Superconvergence of RT0 Mixed Methods for a Class of Semilinear Elliptic Optimal Control Problems

In this paper, we will investigate the error estimates and the superconvergence property of mixed finite element methods for a semilinear elliptic control problem with an integral constraint on control. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element and the control variable is approximated by piecewise constant functions. We derive some superconvergence properties for the control variable and the state variables. Moreover, we derive $L^∞$- and $H^{-1}$-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.