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- W2034797962 abstract "We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta Subscript delta> <mml:semantics> <mml:msub> <mml:mi>β<!-- β --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>beta _{delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the inverse of which enters into error estimates: <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta Subscript delta> <mml:semantics> <mml:msub> <mml:mi>β<!-- β --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>beta _{delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is unity for the heat equation; <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta Subscript delta> <mml:semantics> <mml:msub> <mml:mi>β<!-- β --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>beta _{delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decreases only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time <italic>a posteriori</italic> error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reaction-convection-diffusion equation." @default.
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- W2034797962 date "2013-10-23" @default.
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- W2034797962 title "An improved error bound for reduced basis approximation of linear parabolic problems" @default.
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- W2034797962 doi "https://doi.org/10.1090/s0025-5718-2013-02782-2" @default.
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