A Penalization Method for the Elliptic Bilateral Obstacle Problem

Abstract

International audienceIn this paper we propose a new algorithm for the wellknown elliptic bilateral obstacle problem. Our approach enters the category of fixed domain methods and solves just linear elliptic equations at each iteration. The approximating coincidence set is explicitly computed. In the numerical examples, the algorithm has a fast convergence

Non-convex non-local reactive flows for saliency detection and segmentation

“…The following result generalizes to the non-local framework the results of [17], establishing that the solution of ( 10) is such that the subset of (0, T ) × Ω where u(t, x) / ∈ [0, 1] may be done arbitrarily small by decreasing r. Thus, in the limit r → 0 the solution does not overpass the obstacles u = 0 and u = 1.…”

“…In this section we describe the experiments that support our conclusions. First, we compare the use of hard truncation (in the fully explicit numerical scheme ( 20)) with the iterative scheme (r → 0) when the Yosida's Approximants are used to solve the discrete problem (17). As a second test, the proposed kernel based approach in ( 23) is compared with the patch based scheme in (21).…”

Analysis of a new kind of elastic-rigid bilateral obstacle problem

“…Example 1. We consider this example from the works by Ran and Cheng [1], Murea and Tiba [11], and Cheng et al [18]. Here, ϕ ( x , y ) = − dist ( ( x , y ) , ∂ Ω ) , ψ ( x , y ) = dist ( ( x , y ) , ∂ Ω ) , f ( x , y ) = 11 ( x + y − 1 ) , and let g 1 ( x , y ) = 0 . 001 , g 2 ( x , y ) = 0 . 001 .…”

Numerical analysis for a new kind of obstacle problem

“…The way in which the obstacle problem is constructed has been rigorously described. Assume that the elastic membrane (1) passes through the boundary of a bounded domain ; (2) lies above an obstacle of height ψ ∈ H 1 ( ) with ψ ≤ 0 on ∂ ; and (3) is subject to the action of a vertical force which is proportional to f ∈ L 2 ( ) [15] . Let u be the vertical displacement component of the membrane.…”

“…In the past decades, mathematical theories and numerical analysis of this model, including existence, uniqueness, regularity, numerical methods and convergence results, have been established extensively and thoroughly. Details can be found, e.g., in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]19,25,27] .…”