LOCAL ERROR ESTIMATES FOR SUPG SOLUTIONS OF ADVECTION-DOMINATED ELLIPTIC LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS

We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimi...

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Veröffentlicht in:SIAM journal on numerical analysis Jg. 47; H. 6; S. 4607 - 4638
Hauptverfasser: HEINKENSCHLOSS, MATTHIAS, LEYKEKHMAN, DMITRIY
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2010
Schlagworte:
Adjoints
Advection
Applied mathematics
Boundary layers
Calculus of variations and optimal control
Cauchy Schwarz inequality
Convergence
Environmental pollution
Equations of state
Error rates
Errors
Exact sciences and technology
Finite element method
Greens function
Mathematical analysis
Mathematical models
Mathematics
Methods
Numerical analysis
Numerical analysis. Scientific computation
Optimal control
Optimization
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
65N15
Error estimation
Optimization method
Advection diffusion equation
Partial differential equation
Finite element method
Elliptic equation
49K20
Linear equation
advection-diffusion equations
Convergence rate
65J10
49M25
Boundary layer
Convergence acceleration
streamline upwind/Petrov Galerkin
Initial value problem
Control system
Optimal convergence
Discretization method
local error estimates
Galerkin-Petrov method
Numerical analysis
Boundary value problem
Optimal control
Optimal solution
Optimal approximation
stabilized finite elements
discretization
ISSN:0036-1429, 1095-7170
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Zusammenfassung:We derive local error estimates for the discretization of optimal control problems governed by linear advection-diffusion partial differential equations (PDEs) using the streamline upwind/Petrov Galerkin (SUPG) stabilized finite element method. We show that if the SUPG method is used to solve optimization problems governed by an advection-dominated PDE, the convergence properties of the SUPG method is substantially different from the convergence properties of the SUPG method applied for the solution of an advection-dominated PDE. The reason is that the solution of the optimal control problem involves another advection-dominated PDE, the so-called adjoint equation, whose advection field is just the negative of the advection of the governing PDEs. For the solution of the optimal control problem, a coupled system involving both the original governing PDE as well as the adjoint PDE must be solved. We show that in the presence of a boundary layer, the local error between the solution of the SUPG discretized optimal control problem and the solution of the infinite dimensional problem is only of first order even if the error is computed locally in a region away from the boundary layer. In the presence of interior layers, we prove optimal convergence rates for the local error in a region away from the layer between the solution of the SUPG discretized optimal control problems and the solution of the infinite dimensional problem. Numerical examples are presented to illustrate some of the theoretical results.
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ISSN:0036-1429
1095-7170
DOI:10.1137/090759902