673 research outputs found
An Optimal EDG Method for Distributed Control of Convection Diffusion PDEs
We propose an embedded discontinuous Galerkin (EDG) method to approximate the solution of a distributed control problem governed by convection diffusion PDEs, and obtain optimal a priori error estimates for the state, dual state, their uxes, and the control. Moreover, we prove the optimize-then-discretize (OD) and discrtize-then-optimize (DO) approaches coincide. Numerical results confirm our theoretical results
Error Estimates for Approximations of Distributed Order Time Fractional Diffusion with Nonsmooth Data
In this work, we consider the numerical solution of an initial boundary value
problem for the distributed order time fractional diffusion equation. The model
arises in the mathematical modeling of ultra-slow diffusion processes observed
in some physical problems, whose solution decays only logarithmically as the
time tends to infinity. We develop a space semidiscrete scheme based on the
standard Galerkin finite element method, and establish error estimates optimal
with respect to data regularity in and norms for both smooth
and nonsmooth initial data. Further, we propose two fully discrete schemes,
based on the Laplace transform and convolution quadrature generated by the
backward Euler method, respectively, and provide optimal convergence rates in
the norm, which exhibits exponential convergence and first-order
convergence in time, respectively. Extensive numerical experiments are provided
to verify the error estimates for both smooth and nonsmooth initial data, and
to examine the asymptotic behavior of the solution.Comment: 25 pages, 2 figure
An HDG Method for Distributed Control of Convection Diffusion PDEs
We propose a hybridizable discontinuous Galerkin (HDG) method to approximate
the solution of a distributed optimal control problem governed by an elliptic
convection diffusion PDE. We derive optimal a priori error estimates for the
state, adjoint state, their fluxes, and the optimal control. We present 2D and
3D numerical experiments to illustrate our theoretical results.Comment: arXiv admin note: substantial text overlap with arXiv:1712.10106,
arXiv:1712.01403, arXiv:1712.0293
Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint
In this work, we present numerical analysis for a distributed optimal control
problem, with box constraint on the control, governed by a subdiffusion
equation which involves a fractional derivative of order in
time. The fully discrete scheme is obtained by applying the conforming linear
Galerkin finite element method in space, L1 scheme/backward Euler convolution
quadrature in time, and the control variable by a variational type
discretization. With a space mesh size and time stepsize , we
establish the following order of convergence for the numerical solutions of the
optimal control problem: in the
discrete norm and
in the discrete
norm, with any small and
. The analysis relies essentially on the maximal
-regularity and its discrete analogue for the subdiffusion problem.
Numerical experiments are provided to support the theoretical results.Comment: 20 pages, 6 figure
Mini-Workshop: Numerical Analysis for Non-Smooth PDE-Constrained Optimal Control Problems
This mini-workshop brought together leading experts working on various aspects of numerical analysis for optimal control problems with nonsmoothness. Fifteen extended abstracts summarize the presentations at this mini-workshop
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