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W2072270465 abstract "Consider a solution <italic>u</italic> of the parabolic equation <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u Subscript t Baseline plus upper A u equals f in normal upper Omega times left-bracket 0 comma upper T right-bracket comma> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> </mml:msub> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mspace width=1em /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>in</mml:mtext> </mml:mrow> <mml:mspace width=1em /> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo>×</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{u_t} + Au = fquad {text {in}}quad Omega times [0,T],</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> where <italic>A</italic> is a second order elliptic differential operator. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript h> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{S_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; <italic>h</italic> small denote a family of finite element subspaces of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{H^1}(Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which permits approximation of a smooth function to order <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>. Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega 0 subset-of normal upper Omega> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo>⊂</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{Omega _0} subset Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and assume that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u Subscript h Baseline colon left-bracket 0 comma upper T right-bracket right-arrow upper S Subscript h Baseline> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>:</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo stretchy=false>→</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{u_h}:[0,T] to {S_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an approximate solution which satisfies the semidiscrete interior equation <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis u Subscript h comma t Baseline comma chi right-parenthesis plus upper A left-parenthesis u Subscript h Baseline comma chi right-parenthesis equals left-parenthesis f comma chi right-parenthesis for-all chi element-of upper S Subscript h Superscript 0 Baseline left-parenthesis normal upper Omega 0 right-parenthesis equals StartSet chi element-of upper S Subscript h Baseline comma supp chi subset-of normal upper Omega 0 EndSet comma> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>h</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=1em /> <mml:mi mathvariant=normal>∀</mml:mi> <mml:mi>χ</mml:mi> <mml:mo>∈</mml:mo> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mtext>supp</mml:mtext> </mml:mrow> <mml:mi>χ</mml:mi> <mml:mo>⊂</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:mo fence=false stretchy=false>}</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>({u_{h,t}},chi ) + A({u_h},chi ) = (f,chi )quad forall chi in S_h^0({Omega _0}) = { chi in {S_h},{text {supp}}chi subset {Omega _0}} ,</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 A left-parenthesis dot comma dot right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>A( cdot , cdot )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the bilinear form on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{H^1}(Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated with <italic>A</italic>. It is shown that if the finite element spaces are based on uniform partitions in a specific sense in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega 0> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{Omega _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then difference quotients of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u Subscript h> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{u_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may be used to approximate derivatives of <italic>u</italic> in the interior of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega 0> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{Omega _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to order <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> provided certain weak global error estimates for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=u Subscript h Baseline minus u> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo>−</mml:mo> <mml:mi>u</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>{u_h} - u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to this order are available. This generalizes results proved for elliptic problems by Nitsche and Schatz [9) and Bramble, Nitsche and Schatz [1]." @default.
W2072270465 created "2016-06-24" @default.
W2072270465 creator A5072856113 @default.
W2072270465 date "1979-01-01" @default.
W2072270465 modified "2023-09-23" @default.
W2072270465 title "Some interior estimates for semidiscrete Galerkin approximations for parabolic equations" @default.
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W2072270465 doi "https://doi.org/10.1090/s0025-5718-1979-0514809-4" @default.
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