Problems Of Elliptic Type Research Articles

Existence of Solutions for a Problem of Resonance Using Variational Method

In this work, we used variational methods to show the existence of weak solutions for a nonlinear problem of elliptic type. This problem was first studied by Ahmad et al. (see Indiana Univ Math J, 25, 933–944, 1976) in a bounded domain. In this paper we extend the result to the whole domain \({\mathbb{R}^N}\) .

Partially and globally overdetermined problems of elliptic type

We review some classical and recent results on overdetermined problems both in bounded and unbounded domains, and we discuss how to recover, in some circumstances, a globally overdetermined problem from a partially overdetermined one.

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.

Multi-coefficient Dirichlet Neumann type elliptic inverse problems with application to reflection seismology

Peeter Akerberg Recent Ph.D. graduates are invited to submit abstracts of their dissertations by completing and submitting the submission form 〈http://www.seg.org/SEGportalWEBproject/prod/SEG-Publications/Pub-Geophysics/Documents/dissertationform.pdf〉 at http://seg.org/dissertationabstracts. Abstracts are reviewed and accepted or rejected based on their relevance to the readers of Geophysics. Abstracts must be written in English and defended …

Boundary behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up

By the Karamata regular variation theory and constructing comparison function, we show the exact asymptotic behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up.

Gelfand type elliptic problems under Steklov boundary conditions

For a Gelfand type semilinear elliptic equation we extend some known results for the Dirichlet problem to the Steklov problem. This extension requires some new tools, such as non-optimal Hardy inequalities, and discovers some new phenomena, in particular a different behavior of the branch of solutions and three kinds of blow-up for large solutions in critical growth equations. We also show that small values of the boundary parameter play against strong growth of the nonlinear source.

Multiple solutions to p-Laplacian problems with concave nonlinearities

In this paper we study the p-Laplacian type elliptic problems with concave nonlinearities. Using some asymptotic behavior of f at zero and infinity, three nontrivial solutions are established.

Two-sided a posteriori estimates for a generalized Stokes problem

The paper is concerned with a three-field statement of a generalized Stokes problem related to viscous flow problems for fluids with polymeric chains. For homogeneous Dirichlet boundary conditions, this model and respective numerical methods have been studied previously. In the present paper, a generalized Stokes problem with variable viscosity and nonhomogeneous Dirichlet or mixed Dirichlet/Neumann boundary conditions is considered, and functional a posteriori error estimates for the velocity, pressure, and stress fields are derived. The estimates are practically computable, sharp (i.e., have no gap between the left- and right-hand sides), and are valid for arbitrary functions from respective functional classes. The estimates are obtained by transformations of the integral identity that assigns the generalized solution (this method was suggested and used earlier for certain classes of elliptic type problems). Error majorants are weighted sums of terms penalizing violations of the constitutive, equilibrium, and divergence relations with weights determined by the constants in the Friederichs inequality and the inf-sup (LBB) condition. Bibliography: 53 titles.

Some results on the accuracy of an edge‐based finite volume formulation for the solution of elliptic problems in non‐homogeneous and non‐isotropic media

AbstractThe numerical simulation of elliptic type problems in strongly heterogeneous and anisotropic media represents a great challenge from mathematical and numerical point of views. The simulation of flows in non‐homogeneous and non‐isotropic porous media with full tensor diffusion coefficients, which is a common situation associated with the miscible displacement of contaminants in aquifers and the immiscible and incompressible two‐phase flow of oil and water in petroleum reservoirs, involves the numerical solution of an elliptic type equation in which the diffusion coefficient can be discontinuous, varying orders of magnitude within short distances. In the present work, we present a vertex‐centered edge‐based finite volume method (EBFV) with median dual control volumes built over a primal mesh. This formulation is capable of handling the heterogeneous and anisotropic media using structured or unstructured, triangular or quadrilateral meshes. In the EBFV method, the discretization of the diffusion term is performed using a node‐centered discretization implemented in two loops over the edges of the primary mesh. This formulation guarantees local conservation for problems with discontinuous coefficients, keeping second‐order accuracy for smooth solutions on general triangular and orthogonal quadrilateral meshes. In order to show the convergence behavior of the proposed EBFV procedure, we solve three benchmark problems including full tensor, material heterogeneity and distributed source terms. For these three examples, numerical results compare favorably with others found in literature. A fourth problem, with highly non‐smooth solution, has been included showing that the EBFV needs further improvement to formally guarantee monotonic solutions in such cases. Copyright © 2008 John Wiley & Sons, Ltd.

On the Dirichlet problem of elliptic type

We investigate the existence of solutions and their stability for elliptic Dirichlet problems with nonlinearity of a convex-concave type. By relating the primal action and the dual action functionals on certain subsets of their domains we get the existence of solutions which are further stable with respect to a numerical parameter. We allow also for the differential operator to depend on a numerical parameter.

Error Control in Terms of Linear Functionals Based on Gradient Averaging Techniques

We show how commonly used gradient averaging techniques can be successfully applied to estimation of computational errors evaluated by linear (goal-oriented) functionals for linear elliptic type problems. General scheme for construction of corresponding estimators is described and effectivity of the proposed approach is demonstrated in numerical tests.

Discontinuous Galerkin Methods for Quasi‐Linear Elliptic Problems of Nonmonotone Type

In this paper, both symmetric and nonsymmetric interior penalty discontinuous $hp$‐Galerkin methods are applied to a class of quasi‐linear elliptic problems which are of nonmonotone type. Using Brouwer’s fixed point theorem, it is shown that the discrete problem has a solution, and then, using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in the broken $H^1$‐norm, which are optimal in h and suboptimal in p, are derived. Moreover, on a regular mesh an $hp$‐error estimate for the $L^2$‐norm is also established. Finally, numerical experiments illustrating the theoretical results are provided.

On the accuracy of finite difference methods for elliptic problems with interfaces

In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and by modifying the operator if necessary. It has long been observed that the solution can be computed with uniform O.h 2 / accuracy even if the truncation error is O.h/ at the interface, while O.h 2 / in the interior. We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to O.h 2 log.1=h//. Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Two-fluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is O.h 2 /.

Extending the applicability of multigrid methods

Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements that scale linearly with the degrees of freedom. Unfortunately, they have been historically limited by their applicability to elliptic-type problems and the need for special handling in their implementation. In this paper, we present an overview of several recent theoretical and algorithmic advances made by the TOPS multigrid partners and their collaborators in extending applicability of multigrid methods. specific examples that are presented include quantum chromodynamics, radiation transport, and electromagnetics.

Discrete duality finite volume schemes for Leray−Lions−type elliptic problems on general 2D meshes

AbstractDiscrete duality finite volume schemes on general meshes, introduced by Hermeline and Domelevo and Omnès for the Laplace equation, are proposed for nonlinear diffusion problems in 2D with nonhomogeneous Dirichlet boundary condition. This approach allows the discretization of non linear fluxes in such a way that the discrete operator inherits the key properties of the continuous one. Furthermore, it is well adapted to very general meshes including the case of nonconformal locally refined meshes. We show that the approximate solution exists and is unique, which is not obvious since the scheme is nonlinear. We prove that, for general W−1,p′(Ω) source term and W1‐(1/p),p(∂Ω) boundary data, the approximate solution and its discrete gradient converge strongly towards the exact solution and its gradient, respectively, in appropriate Lebesgue spaces. Finally, error estimates are given in the case where the solution is assumed to be in W2,p(Ω). Numerical examples are given, including those on locally refined meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions

One of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations. For problems of elliptic type, one of the basic properties is the (continuous) maximum principle. In our work, we present several variants of the maximum principles and their discrete counterparts for (scalar) second-order nonlinear elliptic problems with mixed boundary conditions. The problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes. Sufficient conditions providing the validity of the corresponding discrete maximum principles are presented. Geometrically, they mean that the employed meshes have to be of acute or nonobtuse type, depending of the type of the problem. Finally some examples of real-life problems, where the preservation of maximum principles plays an important role, are presented.

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.

On the maximum and comparison principles for a steady‐state nonlinear heat conduction problem

AbstractWe examine a Dirichlet boundary value problem of elliptic type which serves as a model for a stationary heat conduction in nonlinear, inhomogeneous, and anisotropic media. We prove a comparison principle and obtain the maximum principle as a direct consequence. We also show that the standard trilinear finite elements do not preserve a discrete maximum principle.

Laminar Heat Transfer With Viscous Dissipation and Fluid Axial Heat Conduction for Modified Power Law Fluids Flowing in Parallel Plates With One Plate Moving

Using the fully developed laminar velocity distributions obtained by applying the modified power-law model proposed by Irvine and Karni, the thermal-entrance-region heat transfer of non-Newtonian fluids flowing in parallel plates with one plate moving is investigated taking into account both viscous dissipation and fluid axial heat conduction for two kinds of thermal boundary conditions, namely, constant temperature and constant heat flux at the moving wall. The energy equation subject to a constant temperature at upstream infinity, fully developed temperature profile at downstream infinity and the appropriate thermal boundary conditions at the upper and lower walls is numerically solved by the finite difference method as an elliptic type problem. The effects of the moving plate velocity, rheological properties, Brinkman number and Peclet number on the temperature distribution and Nusselt numbers are discussed for both Newtonian and pseudoplastic fluids