Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the Discontinuous Galerkin time discretization

Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh. Exact bounds, that is bounds guaranteed with respect to the exact solution, are needed to properly certify the accuracy of the results, especially if the meshes are coarse. The paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems. In this first part, the bounds account for the error associated with the spatial discretization. The error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error. The time discretization is performed using the discontinuous Galerkin method, both for the primal and adjoint problems. In the error estimation procedure, equilibrated fluxes at interelement edges are calculated using hybridization techniques

A new equilibrated residual method improving accuracy and efficiency of flux-free error estimates

This paper presents a new methodology to compute guaranteed upper bounds for the energy norm of the error in the context of linear finite element approximations of the reaction–diffusion equation. The new approach revisits the ideas in Parés et al. (2009) [6, 4], with the goal of substantially reducing the computational cost of the flux-free method while retaining the good quality of the bounds. The new methodology provides also a technique to compute equilibrated boundary tractions improving the quality of standard equilibration strategies. The zeroth-order equilibration conditions are imposed using an alternative less restrictive form of the first-order equilibration conditions, along with a new efficient minimization criterion. This new equilibration strategy provides much more accurate upper bounds for the energy and requires only doubling the dimension of the local linear systems of equations to be solved.Postprint (author's final draft

Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the discontinuous Galerkin time discretization

Classical implicit residual type error estimators require using an underlying spatial finer mesh to compute bounds for some quantity of interest. Consequently, the bounds obtained are only guaranteed asymptotically that is with respect to the reference solution computed with the fine mesh. Exact bounds, that is bounds guaranteed with respect to the exact solution, are needed to properly certify the accuracy of the results, especially if the meshes are coarse. The paper introduces a procedure to compute strict upper and lower bounds of the error in linear functional outputs of parabolic problems. In this first part, the bounds account for the error associated with the spatial discretization. The error coming from the time marching scheme is therefore assumed to be negligible in front of the spatial error. The time discretization is performed using the discontinuous Galerkin method, both for the primal and adjoint problems. In the error estimation procedure, equilibrated fluxes at interelement edges are calculated using hybridization techniques

Computable exact bounds for linear outputs from stabilized solutions of the advection-diffusion-reaction equation

The paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The bounds are computed using implicit a-posteriori error estimators from stabilized finite element approximations of the exact solution. A new methodology is introduced, based in the ideas presented in [1] for the Galerkin formulation, that allows obtaining bounds also for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the fluxfree technique

A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method

We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced, allowing to derive alternative guaranteed bounds from nearly-arbitrary H(div;{\Omega}) flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements

Computing Bounds for Linear Functionals of Exact Weak Solutions to the Advection-Diffusion-Reaction Equation

We present a cost e#ective method for computing quantitative upper and lower bounds on linear functional outputs of exact weak solutions to the advection-di#usion-reaction equation and we demonstrate a simple adaptive strategy by which such outputs can be computed to a prescribed precision. The bounds are computed from independent local subproblems resulting from a standard finite element approximation to the problem. At the heart of the method lies a local dual problem by which we transform an infinite dimensional minimization problem into a finite dimensional feasibility problem. The bounds hold for all levels of refinement on polygonal domains with piecewise polynomial forcing, and the bound gap converges at twice the rate of the -norm of the error in the finite element solution

Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90641/1/AIAA-53965-537.pd

A-Posteriori bounds on linear functionals of coercive 2nd order PDEs using discontinuous Galerkin methods

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2006.Includes bibliographical references (p. 127-132).In this thesis, we extend current capabilities in producing error bounds on the exact linear functionals of linear partial differential equations in a number of ways. Unlike previous approaches, we base our method on the Discontinuous Galerkin finite element method. For equations such as the convection-diffusion equation, the convection term is handled by the standard DG method for hyperbolic problems while the diffusion operator is discretized by the LDG scheme. This choice allows for the effective bounding of outputs associated with high Peclect number problems without resolving all of the details of the solution. In addition to the ability to manage convection dominated problems, we expand the scope of our error bounding algorithm beyond present capabilities to include saddle problems such as the incompressible Stokes equations. Apart from the aforementioned advantages, the DG discretization employed here also produces associated numerical fluxes, which make the complicated "equilibration" procedure that is often necessary in implicit a-posteriori algorithms, unnecessary.by Joseph S.H. Wong.Ph.D

Asymptotic bounds to outputs of the exact weak solution of the three-dimensional Helmholtz equation

In engineering practice, the design is based on certain design quantities or "outputs of interest" which are functionals of field variables such as displacement, velocity field, or pressure. In order to gain confidence in the numerical approximation of "outputs," a method of obtaining sharp, rigorous upper and lower bounds to outputs of the exact solution have been developed for symmetric and coercive problems (the Poisson equation and the elasticity equation), for non-symmetric coercive problems (advection-diffusion-reaction equation), and more recently for certain constrained problems (Stokes equation). In this thesis we develop the method for the Helmholtz equation. The common approach relies on decomposing the global problem into independent local elemental sub-problems by relaxing the continuity along the edges of a partitioning of the entire domain, using approximate Lagrange multipliers. The method exploits the Lagrangian saddle point property by recasting the output problem as a constrained minimization problem. The upper and lower computed bounds then hold for all levels of refinement and are shown to approach the exact solution at the same rate as its underlying finite element approach. The certificate of precision can then determine the best as well as the worst case scenario in an engineering design problem. This thesis addresses bounds to outputs of interest for the complex Helmholtz equation. The Helmholtz equation is in general non-coercive for high wave numbers and therefore, the previous approaches that relied on duality theory of convex minimization do not directly apply. Only in the asymptotic regime does the Helmholtz equation become coercive, and reliable (guaranteed) bounds can thus be obtained. Therefore, in order to achieve good bounds, several new ingredients have been introduced. The bounds procedure is firstly formulated with appropriate extension to complex-valued equations. Secondly, in the computation of the inter-subdomain continuity multipliers we follow the FETI-H approach and regularize the system matrix with a complex term to make the system non-singular. Finally, in order to obtain sharper output bounds in the presence of pollution errors, higher order nodal spectral element method is employed which has several computational advantages over the traditional finite element approach. We performed verification of our results and demonstrate the bounding properties for the Helmholtz problem

An optimization framework for adaptive higher-order discretizations of partial differential equations on anisotropic simplex meshes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 271-281).Improving the autonomy, efficiency, and reliability of partial differential equation (PDE) solvers has become increasingly important as powerful computers enable engineers to address modern computational challenges that require rapid characterization of the input-output relationship of complex PDE governed processes. This thesis presents work toward development of a versatile PDE solver that accurately predicts engineering quantities of interest to user-prescribed accuracy in a fully automated manner. We develop an anisotropic adaptation framework that works with any localizable error estimate, handles any discretization order, permits arbitrarily oriented anisotropic elements, robustly treats irregular features, and inherits the versatility of the underlying discretization and error estimate. Given a discretization and any localizable error estimate, the framework iterates toward a mesh that minimizes the error for a given number of degrees of freedom by considering a continuous optimization problem of the Riemannian metric field. The adaptation procedure consists of three key steps: sampling of the anisotropic error behavior using element-wise local solves; synthesis of the local errors to construct a surrogate error model based on an affine-invariant metric interpolation framework; and optimization of the surrogate model to drive the mesh toward optimality. The combination of the framework with a discontinuous Galerkin discretization and an a posteriori output error estimate results in a versatile PDE solver for reliable output prediction. The versatility and effectiveness of the adaptive framework are demonstrated in a number of applications. First, the optimality of the method is verified against anisotropic polynomial approximation theory in the context of L2 projection. Second, the behavior of the method is studied in the context of output-based adaptation using advection-diffusion problems with manufactured primal and dual solutions. Third, the framework is applied to the steady-state Euler and Reynolds-averaged Navier-Stokes equations. The results highlight the importance of adaptation for high-order discretizations and demonstrate the robustness and effectiveness of the proposed method in solving complex aerodynamic flows exhibiting a wide range of scales. Fourth, fully-unstructured space-time adaptivity is realized, and its competitiveness is assessed for wave propagation problems. Finally, the framework is applied to enable spatial error control of parametrized PDEs, producing universal optimal meshes applicable for a wide range of parameters.by Masayuki Yano.Ph.D