Boundary Value Problems Of Elliptic Type Research Articles

A note on elliptic type boundary value problems with maximal monotone relations

In this note we discuss an abstract framework for standard boundary value problems in divergence form with maximal monotone relations as “coefficients”. A reformulation of the respective problems is constructed such that they turn out to be unitarily equivalent to inverting a maximal monotone relation in a Hilbert space. The method is based on the idea of “tailor‐made” distributions as provided by the construction of extrapolation spaces, see e.g. [Picard, McGhee: Partial Differential Equations: A unified Hilbert Space Approach (De Gruyter, 2011)]. The abstract framework is illustrated by various examples.

Existence theorems for resonance boundary-value problems of elliptic type with discontinuous unbounded non-linear parts

The existence of a solution of the Dirichlet problem for a second order elliptic equation with non-linear part discontinuous in the phase variable is proved in the cases of resonance on the left and resonance on the right of the first eigenvalue of the differential operator in the situation where the Landesman-Lazer conditions do not hold.

Improving the Ill-conditioning of the Method of Fundamental Solutions for 2D Laplace Equation

The method of fundamental solu- tions (MFS) is a truly meshless numerical method widely used in the elliptic type boundary value problems, of which the approximate solution is expressed as a linear combination of fundamental solutions and the unknown coefficients are deter- mined from the boundary conditions by solving a linear equations system. However, the accuracy of MFS is severely limited by its ill-conditioning of the resulting linear equations system. This paper is motivated by the works of Chen, Wu, Lee and Chen (2007) and Liu (2007a). The first paper proved an equivalent relation of the Tre- fftz method and MFS for circular domain, while the second proposed a modified Trefftz method (MTM). We first prove an equivalent relation of MTM and MFS for arbitrary plane domain. Due to the well-posedness of MTM, we can alleviate the ill-conditioning of MFS through a new linear equations system of the modified MFS (MMFS). In doing so we can raise the accuracy of MMFS over four orders more than the original MFS. Nu- merical examples indicate that the MMFS can attain highly accurate numerical solutions with accuracy over the order of 10 −10 . The present method is fully not similar to the preconditioning technique as used to solve the ill-conditioned lin- ear equations system.

Approximation of the resonance boundary-value problems of elliptic type with a discontinuous nonlinearity

We consider the situation in which a resonance elliptic boundary-value problem with a discontinuous nonlinearity is an idealization of a distributed system with nonlinearities continuous in the phase variable and having narrow areas in the domain of the phase variable in which the tracking of the change of nonlinear parameters is impossible. We study the question of proximity of the solution sets of the original and idealized systems.

Existence Theorem for One Class of Strongly Resonance Boundary-Value Problems of Elliptic Type with Discontinuous Nonlinearities

We consider the Dirichlet problem for an equation of the elliptic type with a nonlinearity discontinuous with respect to the phase variable in the resonance case; it is not required that the nonlinearity satisfy the Landesman-Lazer condition. Using the regularization of the original equation, we establish the existence of a generalized solution of the problem indicated.

A unified approach to a posteriori error estimation based on duality error majorants

In this paper, we present a unified approach to computing sharp error estimates for approximate solutions of elliptic type boundary value problems in variational form. This approach is based on the duality theory of the calculus of variations which analyses the original (primal) variational problem together with another (dual) one. In previous papers this theory was used to obtain a new tool of a posteriori error estimation – duality error majorant (DEM). The latter provides an upper bound for the energy norm of the error for any approximation that belongs to the set of admissible functions of the considered variational problem. For this reason, DEM can be easily used to estimate the accuracy of various post-processed finite element approximations as well as to those computed by boundary element or by finite difference methods. The objective of this paper is to present practically convenient forms of DEM and to discuss computational aspects of this error estimation strategy. The performance of the proposed error estimation method is demonstrated through several examples, where duality majorants are computed and compared with the results obtained by other methods.

Algorithms for the statistical modelling of the solution of elliptic-type boundary value problems

Algorithms for the statistical modelling of the solution of elliptic-type boundary value problems

The boundary integral equation method with application to certain stress concentration problems in elasticity

AbstractThe essential aspects of the Boundary Integral Equation Method for the numerical solution of elliptic type boundary value problems are presented. A numerical example for a stress concentration problem in classical elasticity in three dimensions is given along with several examples for a class of scalar problems in elastic torsion of non-cylindrical bars. Some discussion and criticism of the method itself and in comparison with widely used field methods is also presented.