Value Problems Of Second Order Research Articles

Numerical Enclosure-Methods for the Weak Solution of Elliptic Problems with Unilateral Constraints

In this paper we discuss elliptic boundary value problems of second order with unilateral constraints and give a complete numerical solution scheme related to the lately developed Verification-Methods or E-Methods which are the basis for computations to prove automatically the existence, uniqueness, and an enclosure of the solution. Here we apply the penalty method to prove the existence and uniqueness of the solution of the variational problem and to get finite element approximations which we prove to converge to the solution of the analytic problem. We give the associated bounds for error estimates and then develop direct and iterative E-methods for verified computations.

Existence of bounded solutions for some degenerated quasilinear elliptic equations

We prove the existence of bounded solutions in L ∞ (Ω) of degenerate elliptic boundary value problems of second order in divergence form with natural growth in the gradient. For the Dirichlet problem our results cover also unbounded domains Ω.

Galerkin-wavelet methods for two-point boundary value problems

Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.

On Asymptotics of Solutions of Elliptic Mixed Boundary Value Problems of Second-Order in Domains with Vanishing Edges

This paper investigates the behavior of variational solutions of second-order elliptic mixed boundary value problems (MBVP) with real coefficients in n-dimensional domains with edges near the points where the edges are vanishing. It is shown that the first coefficient involved in the decomposition of the solution into regular and singular part can be extended continuously in appropriate spaces across such points, thus showing that the standard decomposition formula holds also in such domains.

On the stability of index of nonlocal elliptic problems

One of the first publications devoted to nonlocal elliptic problems was the paper of T. Carleman [S]. Together with this paper are closely associated further investigations of singular integral equations with shifts and elliptic boundary value problems with shifts which map the boundary onto itself [9]. Bibliographies of publications and the theory of abstract nonlocal elliptic problems are contained in [4, 71. In [2] the Laplace equation was considered with boundary conditions connecting the values of the unknown function on a manifold r, c aQ (Q c R”) with its values on some manifold o(T, ) c Q and Dirichlet conditions on 13Q\f,. The solvability of this problem in various functional spaces was studied in [16-18,201 and elsewhere. In [ 17,201 it was shown that in the case o(F,) n (r,\r,) # 0 the Dirichlet problem with arbitrary small coefficients in nonlocal terms can have a negative index. In [lo] the formula of index for some nonlocal elliptic boundary value problems of second order in a plane domain was obtained. In the present paper we consider elliptic equations of order 2m in a bounded domain Q c R with nonlocal conditions connecting the traces of the unknown function and its derivatives on the manifolds rj (IJ, F, = 8Q) with its traces on some compacturn a c Q. In the case a c Q (Section 2) and in the case Sz n K = 0 (Section 3) we prove the stability of index of nonlocal problems in Sobolev spaces and weighted spaces, where Xi = Uj (T,\r,). In particular the set D can consist of one or several manifolds (see [2, 163). In Section 4 it is established that in the case Q n Xi = 0 index of the equation of second order, with Dirichlet conditions containing nonlocal perturbations, is equal to zero. The elliptic problems with nonlocal conditions are closely associated with boundary value problems for elliptic functional differential equations [ 171. The theory of functional differential equations for the functions of one variable was reflected in the publications most completely [IS, 131. The 323 0022-247X/91 $3.00

Approximation of solutions for nonlinear second-order boundary value problems

Abstract. The purpose of this paper is to study two classes of nonlinear boundary value problems of second order when the nonlinearity is a Carathéodory function. We develop the monotone method in order to approximate the extremal solutions between the upper and lower solutions.

A numerical approach to the proof of existence of solutions for elliptic problems II

In this paper, we describe a method which proves by computers the existence of weak solutions for linear elliptic boundary value problems of second order. It is shown that we can constitute the computing procedures to verify the existence, uniqueness and inclusion set of a solution based on Schauder’s fixed point theorem. Using the finite element approximations for some simple Poisson’s equations and the results of error estimates, we generate iteratively a set sequence composed of functions and attempt to construct automatically the set including the exact solution. Further, the conditions of verifiability by this method are considered and some numerical examples of verification are presented.

Finite-Element Methods for Singularly Perturbed Elliptic Boundary Value Problems and its Application to the Stationary Navier-Stokes Equations

We consider some modifications of finite-element methods for solving singularly perturbed elliptic beoundary value problems of second order where the reduced problem is of first order. Stability and convergence properties of such methods and the application to the stationary Navier-Stokes problem for high Reynolds numbers are studied.

Boundary value problems for second order singularly perturbed differential equations

In this paper the existence and uniqueness of solutions of boundary value problems (which contains the Robin's problem) is discussed by using the upper and lower solution. In addition, the asymptotic estimation of the solution is given as well.

Collocation for Singular Boundary Value Problems of Second Order

The collocation methods for solving linear boundary value problems with a singularity of the first kind are investigated. The stability and convergence results are established. Particular attention is paid to the superconvergence properties of the methods, which are different as in the classical case when the singularity is not present. The results are extended to the nonlinear case and illustrated by numerical examples.

On the multi-level splitting of finite element spaces

In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log κ)2) where κ is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like $$O\left( {\left( {\log \frac{1}{h}} \right)^2 } \right)$$ instead of $$O\left( {\left( {\frac{1}{h}} \right)^2 } \right)$$ for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.

On the Numerical Solution of Singular Boundary Value Problems of Second Order by a Difference Method

The standard three-point discretization applied to the numerical solution of nonlinear boundary value problems for second-order systems with a singularity of the first kind is investigated. The results are extended to the boundary value problems arising in practical problems from mechanics and chemistry. A number of numerical examples illustrating the theoretical results is presented.

On the numerical solution of singular boundary value problems of second order by a difference method

The standard three-point discretization applied to the numerical solution of nonlinear boundary value problems for second-order systems with a singularity of the first kind is investigated. The results are extended to the boundary value problems arising in practical problems from mechanics and chemistry. A number of numerical examples illustrating the theoretical results is presented.

Quasilineare elliptische Differentialgleichungen zweiter Ordnung in mehrdimensionalen Gebieten mit Kegelspitzen

We study regularity properties for solutions u of a class of quasilinear elliptic boundary value problems of second order in n -dimensional domains with conic vertices. We prove u \in C^{1, \lambda} \cap H^2 near the vertex.

Some Boundary Value Problems for Generalized Differential Equations

Two versions of the Wazewski retract method are proven for generalized differential equations. These theorems are then applied to study some two-point boundary value problems for second order generalized differential equations of the type $x'' \in G( {t,x,x'} )$, where $G( {t,x,x'} )$ is an upper semicontinuous, compact, convex set-valued mapping.

Upper and lower bounds on solutions of initial value problems

It is shown that within the scope of ordinary differential equations, the unknown solutions of linear initial value problems of second order, with arbitrary initial conditions, can be bounded from above and from below in terms of known solutions of especially adapted associated initial value problems. The bounds deduced are shown to be valid in explicitly determined intervals, which depend upon the conditions of the particular problem under consideration.