We propose and analyze the symmetric interior penalty Galerkin (SIPG) finite element method for studying Dirichlet boundary optimal control problem governed by the biharmonic operator. The symmetric property of the mesh dependent bilinear form is a crucial ingredient to derive the discrete optimality system. We perform a priori as well as a posteriori error analysis of the underlying finite element method on any general convex polygonal domain. Under appropriate regularity assumptions, optimal order error bounds are achieved in the energy norm for the control, state and adjoint state. We also derive the error estimate for the control in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2841_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> norm. Moreover, we have derived residual based reliable and efficient a posteriori error estimator for the development of an efficient adaptive algorithm. When the first penalty parameter tends to infinity, the complete fourth order SIPG finite element method reduces to the classic <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2841_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>-interior penalty method. The corresponding error estimates (both a priori and a posteriori) are obtained for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2841_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>-interior penalty method. Numerical results demonstrate the execution of the method and compliments the theoretical findings.

Discontinuous Galerkin Method for Dirichlet Boundary Control Problem Governed by Biharmonic Operator

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发表日期：

2025年3月8日

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Divay Garg,
Kamana Porwal

摘要

We propose and analyze the symmetric interior penalty Galerkin (SIPG) finite element method for studying Dirichlet boundary optimal control problem governed by the biharmonic operator. The symmetric property of the mesh dependent bilinear form is a crucial ingredient to derive the discrete optimality system. We perform a priori as well as a posteriori error analysis of the underlying finite element method on any general convex polygonal domain. Under appropriate regularity assumptions, optimal order error bounds are achieved in the energy norm for the control, state and adjoint state. We also derive the error estimate for the control in $L_2$ $L_{2}$ norm. Moreover, we have derived residual based reliable and efficient a posteriori error estimator for the development of an efficient adaptive algorithm. When the first penalty parameter tends to infinity, the complete fourth order SIPG finite element method reduces to the classic $C^0$ $C^{0}$ -interior penalty method. The corresponding error estimates (both a priori and a posteriori) are obtained for $C^0$ $C^{0}$ -interior penalty method. Numerical results demonstrate the execution of the method and compliments the theoretical findings.