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Anisotropic a posteriori error estimate for an optimal control problem governed by the heat equation
Author(s) -
Picasso Marco
Publication year - 2006
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20156
Subject(s) - mathematics , piecewise , uniqueness , discretization , a priori and a posteriori , partial differential equation , isotropy , polygon mesh , space (punctuation) , optimal control , mathematical analysis , finite element method , mathematical optimization , geometry , computer science , philosophy , physics , epistemology , quantum mechanics , thermodynamics , operating system
The abstract framework of Becker et al. is considered to solve an optimal control problem governed by a parabolic equation. Existence and uniqueness of a solution are proved using the inf‐sup framework and space‐time functional spaces. A Crank‐Nicolson time discretization is proposed, together with continuous, piecewise linear finite elements in space. Existence and uniqueness of a solution to the discretized problem is also proved using the inf‐sup framework. An a posteriori error estimate is proposed, the goal being to control the error between the true and computed cost functional. The error estimate remains valid on strongly anisotropic meshes and an anisotropic error indicator is proposed when the time step is small. Finally, the quality of this error indicator is studied numerically on isotropic and anisotropic meshes. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006