<p>In this study, we will consider the unique continuation problem for reconstructing the final state of the transient Stokes problem when the initial data is unknown, but additional data is given in a subdomain in space-time. The backward differentiation method is used to discretise the time derivative and standard nonconforming affine finite element approximation is applied for the discretisation in space. The discrete system is regularized by adding a penalty of the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1500_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$H^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-semi-norm of the initial data, scaled with the mesh parameter. The scaling is chosen so that an optimal error estimate holds in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1500_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2(T_1,T;H^1(\Omega ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>T</mi> <mo>;</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="211_2025_1500_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$T_1>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The estimate is derived using the Lipschitz stability of the reconstruction problem and interpolation between discrete spaces. The theory is validated on some numerical examples.</p>

A nonconforming finite element method for data assimilation subject to the transient Stokes problem

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2025年10月6日

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Erik Burman,
Deepika Garg,
Janosch Preuss

摘要

In this study, we will consider the unique continuation problem for reconstructing the final state of the transient Stokes problem when the initial data is unknown, but additional data is given in a subdomain in space-time. The backward differentiation method is used to discretise the time derivative and standard nonconforming affine finite element approximation is applied for the discretisation in space. The discrete system is regularized by adding a penalty of the $H^1$ $H^{1}$ -semi-norm of the initial data, scaled with the mesh parameter. The scaling is chosen so that an optimal error estimate holds in $L^2(T_1,T;H^1(\Omega ))$ $L^{2} (T_{1}, T; H^{1} (Ω))$ , $T_1>0$ $T_{1} > 0$ . The estimate is derived using the Lipschitz stability of the reconstruction problem and interpolation between discrete spaces. The theory is validated on some numerical examples.