By Hagander N., Sundblad Y.

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**Extra info for Aufgabensammlung Numerische Methoden. Aufgaben**

**Example text**

Assume ri(i = 1 , 2 ) is an analytic curve of which the origin 0 is a regular point. Let u E C 2 ( R )n C ( R )be a solution of the equation with boundary conditions Assume A, B, C , f , cpl and cp2 are analytic in a neighborhood of the origin and w ( 0 )= cp2 (0) - Then where the regular part U R together with its partial derivatives of all orders remain bounded when x + 0,and with a = T/W, u s ( x )= a) {W O(P r) ln if a # integer, if a = integer. These relations may be formally indefinitely differentiated.

The subscripts "DM, "N" and "C" are intended as shorthand indications for Dirichlet, Neumann and contact boundary conditions. We assume that on rDthe body is clamped, on F N a surface traction of density f E (L2(I'N))d is applied and on Fc the body is in bilateral contact with a rigid foundation. The contact is frictional and is ) ~in R. modeled by Tresca's law. 35) is the equilibrium equation. With a = (aij)dXd, D i v a : R + Ktd is defined by A POSTERIORI ERROR ANALYSIS VIA DUALITY THEORY 34 where the summation convention is used.

Then i f f E WmlP(f2) and m 2 - 2 / p is not an integer, we have the following smoothness property for the solution u (cf, citeGr): for some constants c k , which are certain linear functionals of f . Hence, no matter how smooth the function f is, the smoothness of the solution u is determined by the smoothness of the singular term ul as long as cl = cl ( f ) # 0. We note that ul E w11p(R)if and only if 1 < a 2 / p . An early systematic study of singular behavior around a comer for the solution of an elliptic problem was done by Lehman, cf.