We consider a parabolic–parabolic interface problem and construct a loosely coupled prediction-correction scheme based on the Robin–Robin splitting method analyzed in [J. Numer. Math., 31(1):59–77, 2023]. We show that the errors of the correction step converge at $\mathcal O((\varDelta t)^{2})$, under suitable convergence rate assumptions on the discrete time derivative of the prediction step, where $\varDelta t$ stands for the time-step length. Numerical results are shown to support our analysis and the assumptions.
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Abstract We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system.We prove that the 𝑊-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains.Numerical results are shown for both 𝑊-cycle and 𝑉-cycle algorithms.
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Abstract We investigate a P 1 P_{1} finite element method for an elliptic distributed optimal control problem with pointwise state constraints and a state equation that includes advective/convective and reactive terms.The convergence of this method can be established for general polygonal/polyhedral domains that are not necessarily convex.The discrete problem is a strictly convex quadratic program with box constraints that can be solved efficiently by a primal-dual active set algorithm.more » « less
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We develop multigrid methods for an elliptic distributed optimal control problem on convex domains that are robust with respect to a regularization parameter. We prove the uniform convergence of the $W$-cycle algorithm and demonstrate the performance of $V$-cycle and $W$-cycle algorithms through numerical experiments.more » « less