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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| Published in: | SIAM journal on numerical analysis Vol. 47; no. 6; pp. 4607 - 4638 |
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| Language: | English |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2010
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| ISSN: | 0036-1429, 1095-7170 |
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| Abstract | 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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| AbstractList | 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. |
| Author | HEINKENSCHLOSS, MATTHIAS LEYKEKHMAN, DMITRIY |
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| Keywords | 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 |
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| SubjectTerms | 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 |
| Title | LOCAL ERROR ESTIMATES FOR SUPG SOLUTIONS OF ADVECTION-DOMINATED ELLIPTIC LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS |
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