Fourth-order Elliptic Boundary Value Problems Research Articles

Multiplicity results for some fourth-order elliptic equations with combined nonlinearities

<abstract><p>In this paper, we regard with some fourth-order elliptic boundary value problems involving subcritical polynomial growth and subcritical (critical) exponential growth. Some new existence and multiplicity results are established by using variational methods combined Adams inequality.</p></abstract>

Jacobi–Sobolev Orthogonal Polynomials and Spectral Methods for Elliptic Boundary Value Problems

Generalized Jacobi polynomials with indexes $$\alpha ,\beta \in \mathbb {R}$$ are introduced and some basic properties are established. As examples of applications, the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered, and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems, the Jacobi–Sobolev orthogonal basis functions are constructed, which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.

A new high accuracy method for two-dimensional biharmonic equation with nonlinear third derivative terms: application to Navier–Stokes equations of motion

In this paper, we propose a new compact fourth-order accurate method for solving the two-dimensional fourth-order elliptic boundary value problem with third-order nonlinear derivative terms. We use only 9-point single computational cell in the scheme. The proposed method is then employed to solve Navier–Stokes equations of motion in terms of streamfunction–velocity formulation, and the lid-driven square cavity problem. We describe the derivation of the method in details and also discuss how our streamfunction–velocity formulation is able to handle boundary conditions in terms of normal derivatives. Numerical results show that the proposed method enables us to obtain oscillation-free high accuracy solution.

On fourth-order elliptic boundary value problems with nonmonotone nonlinear function

This paper is concerned with the fourth-order elliptic boundary value problems with nonmonotone nonlinear function. The existence and uniqueness of a solution is proven by the method of upper and lower solutions. A monotone iteration is developed so that the iteration sequence converges monotonically to a maximal solution or a minimal solution, depending on whether the initial iteration is an upper solution or a lower solution.

On efficiency of the dual majorant method for the quality estimation of approximate solutions of fourth-order elliptic boundary value problems

On efficiency of the dual majorant method for the quality estimation of approximate solutions of fourth-order elliptic boundary value problems

On efficiency of the dual majorant method for the quality estimation of approximate solutions of fourth-order elliptic boundary value problems

Using the example of the fourth-order classical boundary value problem, we investigate one of the approaches to the construction of a posteriori estimates of the accuracy of approximate solutions. The emphasis is made on the justification of the effectiveness of the method proposed. The respective conclusions of the theoretical part of the paper are confirmed by a series of numerical experiments.

Block Monotone Iterations for Numerical Solutions of Fourth-Order Nonlinear Elliptic Boundary Value Problems

This paper is concerned with monotone iterative methods for numerical solutions of a class of nonlinear fourth-order elliptic boundary value problems in a two-dimensional domain. The boundary value problem is discretized by the finite difference method, and two iterative processes, called block Jacobi and block Gauss-Seidel monotone iterations, are presented for the computation of solutions of the finite difference system using either an upper solution or a lower solution as the initial iteration. It is shown that the sequence of iterations converges monotonically to a maximal solution or a minimal solution if the nonlinear function is quasi-monotone nondecreasing. A sufficient condition is given to ensure that the maximal and minimal solutions coincide and their common value is the unique solution of the finite difference system. Similar results are obtained for quasi-monotone nonincreasing functions. An analytical comparison relation between the block Jacobi and block Gauss--Seidel monotone iterations is obtained. It is also shown that the finite difference solution converges to the continuous solution as the mesh size tends to zero. Numerical results by the block monotone iterative schemes are given for two model problems and are compared with the known analytical solutions for accuracy. Also compared are the rates of convergence of the two types of block monotone iterations as well as similar types of pointwise monotone iterations.

Numerical methods for fourth‐order nonlinear elliptic boundary value problems

AbstractThe aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001

On fourth-order elliptic boundary value problems

This paper is concerned with the existence and uniqueness of a solution for a class of fourth-order elliptic boundary value problems. The existence of a solution is proven by the method of upper and lower solutions without any monotone nondecreasing or nonincreasing property of the nonlinear function. Sufficient conditions for the uniqueness of a solution and some techniques for the construction of upper and lower solutions are given. All the existence and uniqueness results are directly applicable to fourth-order two-point boundary value problems.

C1-hierarchical bases

In this note we derive estimates for the condition numbers of stiffness matrices relative to certain C 1 conforming hierarchical bases for fourth-order elliptic boundary value problems.

Some iterative Poisson solvers applied to numerical solution of the model fourth-order elliptic problem

The numerical solution of the model fourth-order elliptic boundary value problem in two dimensions is presented. The iterative procedure in which the biharmonic operator is splitted into two Laplace operators is used. After formulating the finite-difference approximation of the procedure, a formula for the evaluation of the transformed iteration vectors is developed. The Jacobi semi-iterative, Richardson and A.D.I. iterative Poisson solvers are applied to compute one transformed iteration vector. By the efficient use of the decomposition property of the corresponding iteration matrices, the fast Fourier transform algorithm needs to be applied twice in the evaluation of one iteration vector. The asymptotic number of operations for the sequential computation is $5n^2 log_2 n$, where $n^2$ is the number of interior grid points in the unit square. The result of$7 log_2 n$ parallel steps for the parallel computation on an SIMD machine with $n^2$ processors is so far the best one.