Error estimates for the approximation of a class of variational inequalities 论文

摘要

In this paper, we prove a general approximation theorem useful in obtaining order of convergence estimates for the approximation of the solutions of a class of variational inequalities. The theorem is then applied to obtain an "optimal" rate of convergence for the approximation of a second-order elliptic problem with convex set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K equals left-brace upsilon element-of upper H 0 Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis colon upsilon greater-than-or-slanted-equals chi"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi> υ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mn>0</mml:mn> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>:</mml:mo> <mml:mi> υ </mml:mi> <mml:mo> ⩾ </mml:mo> <mml:mi> χ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">K = \{ \upsilon \in H_0^1(\Omega ):\upsilon \geqslant \chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a.e. in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> }.