Fourth-order Elliptic Boundary Value Research Articles

Training RBF neural networks for the solution of elliptic boundary value problems

We propose a radial basis function (RBF) neural network method for solving two- and three–dimensional second and fourth order elliptic boundary value problems (BVPs). The neural network in question is trained by minimizing a nonlinear least squares functional, thus determining the optimal values of the various RBF parameters involved. The functional minimization is carried out using standard MATLAB® software efficiently. Several numerical experiments are presented to demonstrate the efficacy of the proposed method.

A C0 Interior Penalty Discontinuous Galerkin Method and an equilibrated a posteriori error estimator for a nonlinear fourth order elliptic boundary value problem of p-biharmonic type

We consider a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of a nonlinear fourth order elliptic boundary value problem of p-biharmonic type and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a discretization of the corresponding minimization problem involving a suitably defined reconstruction operator. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W2,p norm in terms of the associated primal and dual energy functionals. It requires the construction of an equilibrated flux and an equilibrated moment tensor based on a three-field formulation of the C0IPDG approximation. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.

AN APPLICATION OF LINKING THEOREM TO FOURTH ORDER ELLIPTIC BOUNDARY VALUE PROBLEM WITH FULLY NONLINEAR TERM

We show the existence of nontrivial solutions for some fourth order elliptic boundary value problem with fully nonlinear term. We obtain this result by approaching the variational method and using a linking theorem. We also get a uniqueness result.

Global homeomorphism and applications to the existence and uniqueness of solutions of some differential equations

In this paper, the global diffeomorphism theorem due to Zampieri was generalized. The concept of the basin of attraction is the main tool of our exposition in discussing the diffeomorphism between Banach spaces. The existence and uniqueness of a solution of the fourth order elliptic boundary value problem was proved by employing our generalized theorem. The results of this paper generalize some known theorems.MSC:35J40.

Fourth order elliptic boundary value problem with nonlinear term decaying at the origin

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.MSC:35J30, 35J40.

Corners Give Problems When Decoupling Fourth Order Equations Into Second Order Systems

Decoupling a fourth order elliptic boundary value problem into a system of second order equations may give an alternative approach to obtaining a solution for the original problem. The system approach is usually preferred for numerical methods since one may use piecewise linear finite elements to approximate a solution. In this note we compare both settings for the plate equation and prove that for planar domains with corners the system approach may fail to produce the correct solution.

Multigrid Algorithms for C0 Interior Penalty Methods

Multigrid algorithms for $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper. It is shown that $V$-cycle, $F$-cycle and $W$-cycle algorithms are contractions if the number of smoothing steps is sufficiently large. The contraction numbers of these algorithms are bounded by $Cm^{-\alpha}$, where $m$ is the number of presmoothing (and postsmoothing) steps, $\alpha$ is the index of elliptic regularity, and the positive constant $C$ is mesh-independent. These estimates are established for a smoothing scheme that uses a Poisson solve as a preconditioner, which can be easily implemented because the $C^0$ finite element spaces are standard spaces for second order problems. Furthermore the variable $V$-cycle algorithm is also shown to be an optimal preconditioner.

Two-level additive Schwarz preconditioners for C0 interior penalty methods

We study two-level additive Schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting from C0 interior penalty methods for fourth order elliptic boundary value problems. We show that the condition number of the preconditioned system is bounded by C(1+(H3/δ3)), where H is the typical diameter of a subdomain, δ measures the overlap among the subdomains and the positive constant C is independent of the mesh sizes and the number of subdomains.

C 0 Interior Penalty Methods for Fourth Order Elliptic Boundary Value Problems on Polygonal Domains

C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are analyzed in this paper. A post-processing procedure that can generate C1 approximate solutions from the C0 approximate solutions is presented. New C0 interior penalty methods based on the techniques involved in the post-processing procedure are introduced. These new methods are applicable to rough right-hand sides.

Convergence of nonconforming multigrid methods without full elliptic regularity

We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound ( > 1 >1 ) for the contraction number of the W W -cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable V V -cycle algorithm is an optimal preconditioner.

Sign-definite solutions in some linear elliptic systems

SynopsisWe consider a weakly coupled set of two partial differential equations where the coupling matrix has variable elements and the principal part of each equation is the same uniformly elliptic operator. Weobtain necessary conditions that the system of equations can be decoupled. By decoupling the system and using a positivity lemma due to Hess and Kato, we determine the algebraic sign of the solution components. This work extends recent results of de Figueiredo and Mitidieri. Further, one can use these results to determine the sign of the solution to certain fourth order elliptic boundary value problems.

Nondifferentiable Optimization Problems for Elliptic Systems

The paper is devoted to analysis of optimization problems in coefficients of fourth order elliptic boundary value problems. Similar problems were investigated in the framework of shape optimal design of thin plates. Since in general such problems have no optimal solution, G-convergence theory of elliptic operators is exploited in order to define and to characterize generalized optimal solutions. Necessary optimality conditions for nonsmooth optimization problems are derived. Results of computations for two examples are presented.

Existence theorems for some nonlinear fourth order elliptic boundary value problems

The general outline of our method is to that of of the second author second order elliptic parabolic equations [S, 71. of our effort is in deriving estimates for linear equations which lead to for the FrCchet derivatives of the relevant nonlinear to which Section 2 devoted, appear to be sharp in to explained below, and feel are independent interest. are stated in and the existence theorems developed in 4. a concluding section, a comparison is made between our results and on the theory of and pseudomonotone operators. There is substantial overlap in classes of to which the results and those obtained by The latter generally allows much greater in the nonlinearity (without priori estimate being available) whereas our results allow for a certain amount of ‘nonmonotonicity’ in the nonlinear terms. Furthermore, here we obtain a constructive algorithm (a variation of Newton’s method) for the solution, and this is a large part of the motivation for this work. In fact, the second author has recently utilized the second order results in [5] to obtain numerical approximations by the finite element method in [2]. We anticipate that the present work can be similarly utilized.

Approximation of the Boundary in the Finite Element Solution of Fourth Order Problems

A procedure is presented to handle curved boundaries in the finite element solution of fourth order elliptic boundary value problems over bounded regions $\Omega $ of the plane. The region $\Omega $ is triangulated with triangles with at most one curved side along the boundary. Finite elements are constructed on the curved boundary triangles which can be pieced together with conforming finite elements on interior triangles to obtain trial functions which are globally in $C^1$. Error estimates are derived in the energy norm and the effect of numerical integration is analyzed.

On Solving Nonlinear Equations with a One-Parameter Operator Imbedding

One parameter operator imbedding to modify Newton method for solution of nonlinear equations