Explicit-in-time variational formulations for goal-oriented adaptivity

Abstract

Goal-Oriented Adaptivity (GOA) is a powerful tool to accurately approximate physically relevantfeatures of the solution of Partial Differential Equations (PDEs). It delivers optimal grids to solvechallenging engineering problems. In time dependent problems, GOA requires to represent the error inthe Quantity of Interest (QoI) as an integral over the whole space-time domain in order to reduce it viaadaptive refinements. A full space-time variational formulation of the problem allows theaforementioned error representation. Thus, variational space-time formulations for PDEs have been ofgreat interest in the last decades, among other things, because they allow to develop mesh-adaptivealgorithms. Since it is known that implicit time marching schemes have variational structure, they areoften employed for GOA in time-domain problems. When coming to explicit-in-time methods, thesewere introduced for Ordinary Differential Equations (ODEs) employing specific inexact quadrature rules.In this dissertation, we prove that the explicit Runge-Kutta (RK) methods can be expressed asdiscontinuous-in-time Petrov-Galerkin (dPG) methods for the linear advection-diffusion equation. Wesystematically build trial and test functions that, after exact integration in time, lead to one, two, andgeneral stage explicit RK methods. This approach enables us to reproduce the existing time-domain goalorientedadaptive algorithms using explicit methods in time. Here, we employ the lowest order dPGformulation that we propose to recover the Forward Euler method and we derive an appropriate errorrepresentation. Then, we propose an explicit-in-time goal-oriented adaptive algorithm that performs localrefinements in space. In terms of time domain adaptivity, we impose the Courant-Friedrichs-Lewy (CFL)condition to ensure the stability of the method. We provide some numerical results in one-dimensional(1D)+time for the diffusion and advection-diffusion equations to show the performance of the proposedalgorithm.On the other hand, time-domain adaptive algorithms involve solving a dual problem that runs backwardsin time. This process is, in general, computationally expensive in terms of memory storage. In this work,we define a pseudo-dual problem that runs forwards in time. We also describe a forward-in-time adaptivealgorithm that works for some specific problems. Although it is not possible to define a general dualproblem running forwards in time that provides information about future states, we provide numericalevidence via one-dimensional problems in space to illustrate the efficiency of our algorithm as well as itslimitations. As a complementary method, we propose a hybrid algorithm that employs the classicalbackward-in-time dual problem once and then performs the adaptive process forwards in time. We alsogeneralize a novel error representation for goal-oriented adaptivity using (unconventional) pseudo-dualproblems in the context of frequency-domain wave-propagation problems to the time-dependent waveequation. We show via 1D+time numerical results that the upper bounds for the new error representationare sharper than the classical ones. Therefore, this new error representation can be used to design moreefficient goal-oriented adaptive methodologies.Finally, as classical Galerkin methods may lead to instabilities in advection-dominated-diffusionproblems and therefore, inappropriate refinements, we propose a novel stabilized discretization method,which we call Isogeometric Residual Minimization (iGRM) with direction splitting. This methodcombines the benefits resulting from Isogeometric Analysis (IGA), residual minimization, andAlternating Direction Implicit (ADI) methods. We employ second order ADI time integrator schemes, Bsplinebasis functions in space and, at each time step, we solve a stabilized mixed method based onresidual minimization. We show that the resulting system of linear equations has a Kronecker productstructure, which results in a linear computational cost of the direct solver, even using implicit timeintegration schemes together with the stabilized mixed formulation. We test our method in 2D and3D+time advection-diffusion problems. The derivation of a time-domain goal-oriented strategy based oniGRM will be considered in future works.