A posteriori error estimation for variational problems with uniformly convex functionals

Abstract

The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf Underscript v element-of upper V Endscripts left-brace upper F left-parenthesis v right-parenthesis plus upper G left-parenthesis normal upper Lamda v right-parenthesis right-brace comma"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo form="prefix">inf</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>v</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:munder> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\inf \limits _{v\in V} \{ F(v)+G(\Lambda v) \},</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F colon upper V right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>:</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">F:V\rightarrow \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a convex lower semicontinuous functional, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G colon upper Y right-arrow double-struck upper R"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">G: Y\rightarrow \mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a uniformly convex functional, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are reflexive Banach spaces, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda colon upper V right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Λ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda :V\rightarrow Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the <italic>duality error estimate</italic> obtained and, thus, can be justified via the duality theory.