Abstract
We derive space-time local a posteriori error estimates of finite element approximations to Neumann boundary control problems governed by parabolic partial differential equations in a convex bounded domain \(\varOmega \subset {\mathbb {R}}^d\; (d\le 3)\) with Lipschitz boundary. We approximate the state and co-state variables by using piecewise linear and continuous finite elements whereas piecewise constant functions are used to approximate the control variable. The time-discretization is based on the backward Euler method. We derive three different reliable local a posteriori error estimates for Neumann boundary control problems with the observations of the boundary state, the distributed state and the final state. Our derived estimators are of local character in the sense that the leading terms of the estimators depend on the small neighbourhood of the boundary. These new local a posteriori error bounds can be used to study the behaviour of the state and co-state variables around the boundary and provide the necessary feedback in terms of the error indicators for the adaptive mesh refinements in the finite element method. Our numerical results validate the effectiveness of the derived estimators.
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The authors are grateful to the anonymous reviewer for the valuable suggestions which greatly improve the content of this manuscript.
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Manohar, R., Sinha, R.K. Local a Posteriori Error Analysis of Finite Element Method for Parabolic Boundary Control Problems. J Sci Comput 91, 17 (2022). https://doi.org/10.1007/s10915-022-01788-w
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DOI: https://doi.org/10.1007/s10915-022-01788-w
Keywords
- Neumann boundary control problems
- Parabolic partial differential equations
- Finite element method
- The backward-Euler method
- Local a posteriori error estimates