Biharmonic Obstacle Problem: Guaranteed and Computable Error Bounds for Approximate Solutions
- PDF / 1,605,486 Bytes
- 16 Pages / 612 x 792 pts (letter) Page_size
- 35 Downloads / 206 Views
L DIFFERENTIAL EQUATIONS
Biharmonic Obstacle Problem: Guaranteed and Computable Error Bounds for Approximate Solutions D. E. Apushkinskayaa,b,* and S. I. Repinc,d,** a
Saarland University, P.O. Box 151150, Saarbrücken, 66041 Germany Peoples’Friendship University of Russia (RUDN University), Moscow, 117198 Russia cSteklov Institute of Mathematics at St. Petersburg, St. Petersburg, 191023 Russia d University of Jyväskylä, P.O. Box 35 (Agora), Jyväskylä, 40014 Finland *e-mail: [email protected] **e-mail: [email protected]
b
Received June 17, 2020; revised June 17, 2020; accepted July 7, 2020
Abstract—The paper is concerned with an elliptic variational inequality associated with a free boundary obstacle problem for the biharmonic operator. We study the bounds of the difference between the exact solution (minimizer) of the corresponding variational problem and any function (approximation) from the energy class satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle. Using the general theory developed for a wide class of convex variational problems we deduce the error identity. One part of this identity characterizes the deviation of the function (approximation) from the exact solution, whereas the other is a fully computed value (it depends only on the data of the problem and known functions). In real life computations, this identity can be used to control the accuracy of approximate solutions. The measure of deviation from the exact solution used in the error identity contains terms of different nature. Two of them are the norms of the difference between the exact solutions (of the direct and dual variational problems) and corresponding approximations. Two others are not representable as norms. These are nonlinear measures vanishing if the coincidence set defined by means of an approximate solution satisfies certain conditions (for example, coincides with the exact coincidence set). The error identity is true for any admissible (conforming) approximations of the direct variable, but it imposes some restrictions on the dual variable. We show that these restrictions can be removed, but in this case the identity is replaced by an inequality. For any approximations of the direct and dual variational problems, the latter gives an explicitly computable majorant of the deviation from the exact solution. Several examples illustrating the established identities and inequalities are presented. Keywords: variational inequalities, estimates of the distance to the exact solution, aposteriori estimates DOI: 10.1134/S0965542520110032
1. INTRODUCTION Let Ω be a bounded domain in Rd with Lipschitz continuous boundary ∂Ω . By ν we denote an outward unit normal to ∂Ω , and ϕ denotes a given function (obstacle) in C 2(Ω) such that ϕ ≤ 0 on ∂Ω . Throughout the paper, we use the standard notation for the Lebesgue and Sobolev spaces of functions. By g⊕ we denote max{g,0}. Consider the variational problem (3 ): minimize the functional
J (v ) =
( 2 Δv 1
2
Ω
over the closed convex set
)
− f
Data Loading...
