Search results for "Obstacle problem"

Thin obstacle problem : Estimates of the distance to the exact solution

2018

We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution and any function that satisfies the boundary condition and is admissible with respect to the obstacle condition (i.e., they are valid for any approximation regardless of the method by which it was found). Computation of the estimates does not require knowledge of the exact solution and uses only the problem data and an approximation. The estimates provide guaranteed upper bounds of the error (error majorants) and vanish if and only if the approximation c…

Numerical Algorithms Based on Characteristic Domain Decomposition for Obstacle Problems

1997

A new numerical solution algorithm for obstacle problems is proposed, where the characteristic domain decomposition into active and inactive subdomains separated by the free boundary is approximated by a Schwarz method. Such an approach gives an opportunity to apply fast linear system solvers to genuinely non-linear obstacle problems. Other solution algorithms, like projected relaxation methods and active set strategies, are compared to the new solution algorithm. Numerical experiments related to the elastoplastic torsion problem are included showing the efficiency of the new approach.

Mass transport problems for the Euclidean distance obtained as limits of p-Laplacian type problems with obstacles

2014

In this paper we analyze a mass transportation problem that consists in moving optimally (paying a transport cost given by the Euclidean distance) an amount of a commodity larger than or equal to a fixed one to fulfil a demand also larger than or equal to a fixed one, with the obligation of paying an extra cost of −g1(x) for extra production of one unit at location x and an extra cost of g2(y) for creating one unit of demand at y. The extra amounts of mass (commodity/demand) are unknowns of the problem. Our approach to this problem is by taking the limit as p→∞ to a double obstacle problem (with obstacles g1, g2) for the p-Laplacian. In fact, under a certain natural constraint on the extra …

The Obstacle Problem in a Non-Linear Potential Theory

1988

M. Brelot gave rise to the concept harmonic space when he extended classical potential theory on ℝn to an axiomatic system on a locally compact space. I have recently constructed1 a non-linear harmonic space by dropping the assumption that the sum of two harmonic functions is harmonic and considering some other axioms instead. This approach has its origin in the work of O. Martio, P. Lindqvist and S. Granlund2,3,4, who have developed a non-linear potential theory on ℝn connected with variational integrals of the type ∫ F(x,∇u(x)) dm(x), where F(x, h) ≈ |h|p.

A variational inequality approach to the problem of the design of the optimal covering of an obstacle

2005

A Projected Algebraic Multigrid Method for Linear Complementarity Problems

2011

We present an algebraic version of an iterative multigrid method for obstacle problems, called projected algebraic multigrid (PAMG) here. We show that classical AMG algorithms can easily be extended to deal with this kind of problem. This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising, for example, in financial engineering.

A Boundary Control Approach to an Optimal Shape Design Problem

1989

Abstract We consider the problem of controlling the coincidence set in connection with an obstacle problem. We shall transform the obtained optimal shape design problem into a boundary control problem with Dirichlet boundary conditions.

Electrostatic backscattering by insulating obstacles

2012

AbstractWe introduce and analyze backscattering data for a three-dimensional obstacle problem in electrostatics. In particular, we investigate the asymptotic behavior of these data as (i) the measurement point goes to infinity and (ii) the obstacles shrink to individual points. We also provide numerical simulations of these data.

OBSTACLE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS WITH NONHOMOGENEOUS NONLINEAR BOUNDARY CONDITIONS

2008

In this paper we study the questions of existence and uniqueness of solutions for equations of type - div a(x,Du) + γ(u) ∋ ϕ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x,Du) · η + β(u) ∋ ψ. The nonlinear elliptic operator div a(x,Du) modeled on the p-Laplacian operator Δp(u) = div (|Du|p-2Du), with p > 1, γ and β maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) ∩ β(0), [Formula: see text] and the data ϕ ∈ L1(Ω) and ψ ∈ L1(∂ Ω). Since D(γ) ≠ ℝ, we are dealing with obstacle problems. For this kind of problems the existence of weak solution, in the usual sense, fails to be true for nonhomogeneous boundary conditions, so a new concept of solut…

Nonlinear balayage on metric spaces

2009

We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and p-harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage. Original Publication:Anders Björn, Jana Björn, Tero Mäkäläinen and Mikko Parviainen, Nonlinear balayage on metric spaces, 2009, Nonlinear Analysis, (71), 5-6, 2153-2171.http://dx.doi.org/10.1016/j.na.2009.01.051Copyright: Elsevier Science B.V., Amsterdam.http://www.elsevier.com/