Error estimates for semidiscrete finite element methods for parabolic integro-differential equations

Abstract

The purpose of this paper is to attempt to carry over known results for spatially discrete finite element methods for linear parabolic equations to integro-differential equations of parabolic type with an integral kernel consisting of a partial differential operator of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta less-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta \leq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It is shown first that this is possible without restrictions when the exact solution is smooth. In the case of a homogeneous equation with nonsmooth initial data <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v comma v element-of upper L 2"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">v,v \in {L_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , optimal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis h Superscript r Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>h</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O({h^r})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> convergence for positive time is possible in general only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r less-than-or-equal-to 4 minus beta"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>4</mml:mn> <mml:mo> − </mml:mo> <mml:mi> β </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">r \leq 4 - \beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This depends on the fact that the exact solution is then only in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 4 minus beta"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> <mml:mo> − </mml:mo> <mml:mi> β </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^{4 - \beta }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .