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W2091275824 abstract "Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartFraction partial-differential u Over partial-differential t EndFraction minus sigma-summation Underscript i equals 1 Overscript upper N Endscripts StartFraction partial-differential Over partial-differential x Subscript i Baseline EndFraction left-parenthesis StartAbsoluteValue u EndAbsoluteValue Superscript rho minus 2 Baseline StartFraction partial-differential u Over partial-differential x Subscript i Baseline EndFraction right-parenthesis equals f left-parenthesis x comma t right-parenthesis> <mml:semantics> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi mathvariant=normal>∂</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant=normal>∂</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:mfrac> <mml:mo>−</mml:mo> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:munderover> <mml:mfrac> <mml:mi mathvariant=normal>∂</mml:mi> <mml:mrow> <mml:mi mathvariant=normal>∂</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ρ</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mfrac> <mml:mrow> <mml:mi mathvariant=normal>∂</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant=normal>∂</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>frac {partial u}{partial t}-sum ^N_{i=1}frac partial {partial x_i}(|u|^{rho -2}frac {partial u}{ partial x_i})=f(x,t)</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> on bounded convex domains are considered. The range of the parameter <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho> <mml:semantics> <mml:mi>ρ</mml:mi> <mml:annotation encoding=application/x-tex>rho</mml:annotation> </mml:semantics> </mml:math> </inline-formula> includes the fast diffusion case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 greater-than rho greater-than 2> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>1>rho >2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L Superscript rho Baseline left-parenthesis normal upper Omega right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant=normal>∞</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>ρ</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^infty (0,T;L^rho (Omega ))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm with an error controlled by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis normal upper Delta t Superscript one fourth Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(Delta t^{frac 14})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 greater-than rho greater-than 2> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>1>rho >2</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 O left-parenthesis normal upper Delta t Superscript StartFraction 1 Over 2 rho EndFraction Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>ρ</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(Delta t^{frac 1{2rho }})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 less-than-or-equal-to rho greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>2le rho >infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For the fully discrete problem, a global convergence rate of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis normal upper Delta t Superscript one fourth Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(Delta t^{frac 14})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L squared left-parenthesis 0 comma upper T semicolon upper L Superscript rho Baseline left-parenthesis normal upper Omega right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>ρ</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^2(0,T;L^rho (Omega ))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm is shown for the range <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartFraction 2 upper N Over upper N plus 1 EndFraction greater-than rho greater-than 2> <mml:semantics> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> <mml:mo>></mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>frac {2N}{N+1}>rho >2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 less-than-or-equal-to rho greater-than normal infinity> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>></mml:mo> <mml:mi mathvariant=normal>∞</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>2le rho >infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a rate of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis normal upper Delta t Superscript StartFraction 1 Over 2 rho EndFraction Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>ρ</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O(Delta t^{frac 1{2rho }})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is shown in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper L Superscript rho Baseline left-parenthesis 0 comma upper T semicolon upper L Superscript rho Baseline left-parenthesis normal upper Omega right-parenthesis right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>ρ</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>ρ</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Ω</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>L^rho (0,T;L^rho (Omega ))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> norm." @default.
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W2091275824 date "1999-02-11" @default.
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W2091275824 title "A priori $L^rho$ error estimates for Galerkin approximations to porous medium and fast diffusion equations" @default.
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