Explicit-in-Time Variational Formulations for Goal-Oriented Adaptivity

Goal-Oriented Adaptivity (GOA) is a powerful tool to accurately approximate physically relevant features of the solution of Partial Differential Equations (PDEs). It delivers optimal grids to solve challenging engineering problems. In time dependent problems, GOA requires to represent the error in the Quantity of Interest (QoI) as an integral over the whole space-time domain in order to reduce it via adaptive refinements. A full space-time variational formulation of the problem allows the aforementioned error representation. Thus, variational spacetime formulations for PDEs have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes have variational structure, they are often employed for GOA in time-domain problems. When coming to explicit-intime methods, these were introduced for Ordinary In this dissertation, we prove that the explicit Runge-Kutta (RK) methods can be expressed as discontinuous-in-time Petrov-Galerkin (dPG) methods for the linear advection-diffusion equation. We systematically build trial and test functions that, after exact integration in time, lead to one, two, and general stage explicit RK methods. This approach enables us to reproduce the existing time domain goal-oriented adaptive algorithms using explicit methods in time. Here, we employ the lowest order dPG formulation that we propose to recover the Forward Euler method and we derive an appropriate error representation. Then, we propose an explicit-in-time goal-oriented adaptive algorithm that performs local refinements 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 proposed algorithm. On the other hand, time-domain adaptive algorithms involve solving a dual problem that runs backwards in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we dene a pseudo-dual problem that runs forwards in time. We also describe a forward-in-time adaptive algorithm that works for some specific problems. Although it is not possible to dene a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one-dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. As a complementary method, we propose a hybrid algorithm that employs the classical backward-in-time dual problem once and then performs the adaptive process forwards in time. We also generalize a novel error representation for goal-oriented adaptivity using (unconventional) pseudo-dual problems in the context of frequency-domain wave-propagation problems to the time-dependent wave equation. We show via 1D+time numerical results that the upper bounds for the new error representation are sharper than the classical ones. Therefore, this new error representation can be used to design more efficient goal-oriented adaptive methodologies. Finally, as classical Galerkin methods may lead to instabilities in advection-dominated-diffusion problems and therefore, inappropriate refinements, we propose a novel stabilized discretization method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. This method combines the benefits resulting from Isogeometric Analysis (IGA), residual minimization, and Alternating Direction Implicit (ADI) methods. We employ second order ADI time integrator schemes, B-spline basis functions in space and, at each time step, we solve a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method in 2D and 3D+time advection-diffusion problems. The derivation of a time-domain goal-oriented strategy based on iGRM will be considered in future works

Exploiting Kronecker structure in exponential integrators: Fast approximation of the action of phi-functions of matrices via quadrature

In this article, we propose an algorithm for approximating the action of  $\varphi$-functions of matrices against vectors, which is a key operation in exponential time integrators. In particular, we consider matrices with Kronecker sum structure, which arise from problems admitting a tensor product representation. The method is based on quadrature approximations of the integral form of the  $\varphi$-functions combined with a scaling and modified squaring method. Owing to the Kronecker sum representation, only actions of 1D matrix exponentials are needed at each quadrature node and assembly of the full matrix can be avoided. Additionally, we derive a priori bounds for the quadrature error, which show that, as expected by classical theory, the rate of convergence of our method is supergeometric. Guided by our analysis, we construct a fast and robust method for estimating the optimal scaling factor and number of quadrature nodes that minimizes the total cost for a prescribed error tolerance. We investigate the performance of our algorithm by solving several linear and semilinear time-dependent problems in 2D and 3D. The results show that our method is accurate and orders of magnitude faster than the current state-of-the-art

Equivalence between the DPG method and the Exponential Integrators for linear parabolic problems

The Discontinuous Petrov-Galerkin (DPG) method and the exponential integrators are two well established numerical methods for solving Partial Di fferential Equations (PDEs) and sti ff systems of Ordinary Di fferential Equations (ODEs), respectively. In this work, weapply the DPG method in the time variable for linear parabolic problems and we calculate the optimal test functions analytically. We show that the DPG method in time is equivalent to exponential integrators for the trace variables, which are decoupled from the interior variables. In addition, the DPG optimal test functions allow us to compute the approximated solutions in the time element interiors. This DPG method in time allows to construct a posteriori error estimations in order to perform adaptivity. We generalize this novel DPG-based time-marching scheme to general fi rst order linear systems of ODEs. We show the performance of the proposed method for 1D and 2D + time linear parabolic PDEs after discretizing in space by the nite element method

A DPG-based time-marching scheme for linear hyperbolic problems

The Discontinuous Petrov-Galerkin (DPG) method is a widely employed discretization method for Partial Di fferential Equations (PDEs). In a recent work, we applied the DPG method with optimal test functions for the time integration of transient parabolic PDEs. We showed that the resulting DPG-based time-marching scheme is equivalent to exponential integrators for the trace variables. In this work, we extend the aforementioned method to time-dependent hyperbolic PDEs. For that, we reduce the second order system in time to first order and we calculate the optimal testing analytically. We also relate our method with exponential integrators of Gautschi-type. Finally, we validate our method for 1D/2D + time linear wave equation after semidiscretization in space with a standard Bubnov-Galerkin method. The presented DPG-based time integrator provides expressions for the solution in the element interiors in addition to those on the traces. This allows to design di fferent error estimators to perform adaptivity

The DPG Method for the Convection-Reaction Problem, Revisited

We study both conforming and non-conforming versions of the practical DPG method for the convection-reaction problem. We determine that the most common approach for DPG stability analysis - construction of a local Fortin operator - is infeasible for the convection-reaction problem. We then develop a line of argument based on a direct proof of discrete stability; we find that employing a polynomial enrichment for the test space does not suffice for this purpose, motivating the introduction of a (two-element) subgrid mesh. The argument combines mathematical analysis with numerical experiments

Error representation of the time-marching DPG scheme

In this article, we introduce an error representation function to perform adaptivity in time of the recently developed time-marching Discontinuous Petrov–Galerkin (DPG) scheme. We first provide an analytical expression for the error that is the Riesz representation of the residual. Then, we approximate the error by enriching the test space in such a way that it contains the optimal test functions. The local error contributions can be efficiently computed by adding a few equations to the time-marching scheme. We analyze the quality of such approximation by constructing a Fortin operator and providing an a posteriori error estimate. The time-marching scheme proposed in this article provides an optimal solution along with a set of efficient and reliable local error contributions to perform adaptivity. We validate our method for both parabolic and hyperbolic problems

Exploiting the Kronecker product structure of φ−functions in exponential integrators

Exponential time integrators are well-established discretization methods for time semilinear systems of ordinary differential equations. These methods use (Formula presented.) functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the (Formula presented.) function action over vectors for two-dimensional (2D) matrices expressed as a Kronecker sum. For that, we present an auxiliary exponential-related matrix function that we express using Kronecker products of one-dimensional matrices. We exploit state-of-the-art implementations of (Formula presented.) functions to compute this auxiliary function's action and then recover the original (Formula presented.) action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen–Cahn equation

Combining DPG in space with DPG time-marching scheme for the transient advection–reaction equation

In this article, we present a general methodology to combine the Discontinuous Petrov-Galerkin (DPG) method in space and time in the context of methods of lines for transient advection-reaction problems. We  rst introduce a semidiscretization in space with a DPG method rede ning the ideas of optimal testing and practicality of the method in this context. Then, we apply the recently developed DPG-based time-marching scheme, which is of exponential-type, to the resulting system of Ordinary Differential Equations (ODEs). We also discuss how to e ciently compute the action of the exponential of the matrix coming from the space semidiscretization without assembling the full matrix. Finally, we verify the proposed method for 1D+time advection-reaction problems showing optimal convergence rates for smooth solutions and more stable results for linear conservation laws comparing to the classical exponential integrators

Variational Formulations for Explicit Runge-Kutta Methods

Variational space-time formulations for partial di fferential equations have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes have variational structure, they are often employed for adaptivity. Previously, Galerkin formulations of explicit methods were introduced for ordinary di fferential equations employing speci fic inexact quadrature rules. In this work, we prove that the explicit Runge-Kutta methods can be expressed as discontinuous-in-time Petrov-Galerkin methods for the linear di ffusion equation. We systematically build trial and test functions that, after exact integration in time, lead to one, two, and general stage explicit Runge-Kutta methods. This approach enables us to reproduce the existing time-domain (goal-oriented) adaptive algorithms using explicit methods in time

Forward-in-Time Goal-Oriented Adaptivity

In goal-oriented adaptive algorithms for partial differential equations, we adapt the finite element mesh in order to reduce the error of the solution in some quantity of interest. In time-dependent problems, this adaptive algorithm involves solving a dual problem that runs backward in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we define a pseudo-dual problem that runs forward in time. We also describe a forward-in-time adaptive algorithm that works for some specific problems. Although it is not possible to define a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one-dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. Finally, we propose a hybrid algorithm that employs the classical backward-in-time dual problem once and then performs the adaptive process forwards in time