Adjoint Consistency Research Articles

An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

In this paper we present an arbitrary-order fully discrete Stokes complex on general polyhedral meshes. We enrich the fully discrete de Rham complex with the addition of a full gradient operator defined on vector fields and fitting into the complex. We show a complete set of results on the novelties of this complex: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. The Stokes complex is especially well suited for problem involving Jacobian, divergence and curl, like the Stokes problem or magnetohydrodynamic systems. The framework developed here eases the design and analysis of schemes for such problems. Schemes built that way are nonconforming and benefit from the exactness of the complex. We illustrate with the design and study of a scheme solving the Stokes equations and validate the convergence rates with various numerical tests.

Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors

We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.

An interface penalty parameter free nonconforming cut finite element method for elliptic interface problems

In this paper, we explore a nonconforming cut finite element method based on the nonsymmetric version of Nitsche's method without interface penalty for elliptic interface problems. In contrast to symmetric Nitsche methods, this penalty parameter free scheme does not need “sufficiently large” parameters to ensure the stability. Our main results are that a special function is constructed and the stability of the bilinear form is proved by an inf-sup argument. The optimal convergence in the energy norm and the suboptimal convergence in L2-norm are derived. The suboptimality is due to the lack of adjoint consistency of our formulation. Numerical examples are provided to support the theoretical analysis.

Analysis of finite-volume discrete adjoint fields for two-dimensional compressible Euler flows

This work deals with a number of questions related to the discrete and continuous adjoint fields associated with the compressible Euler equations and classical aerodynamic functions. The consistency of the discrete adjoint equations with the corresponding continuous adjoint partial differential equations is one of them. It has been established or at least discussed only for some numerical schemes and a contribution of this article is to give the adjoint consistency conditions for the 2-D Jameson-Schmidt-Turkel scheme in cell-centered finite-volume formulation. The consistency issue is also studied here from a new heuristic point of view by discretizing the continuous adjoint equation for the discrete flow and adjoint fields. Both points of view prove to provide useful information. Besides, it has been often noted that discrete or continuous inviscid lift and drag adjoint exhibit numerical divergence close to the wall and stagnation streamline for a wide range of subsonic and transonic flow conditions. This is analyzed here using the physical source term perturbation method introduced in reference [1]. With this point of view, the fourth physical source term of [1] appears to be the only one responsible for this behavior. It is also demonstrated that the numerical divergence of the adjoint variables corresponds to the response of the flow to the convected increment of stagnation pressure and diminution of entropy created at the source and the resulting change in lift and drag.

An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincaré Inequalities, and Consistency

In this paper, we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in the Koszul complex. The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of text {L}^2-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart–Thomas–Nédélec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincaré inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.

Goal-oriented mesh adaptation method for nonlinear problems including algebraic errors

We deal with the goal-oriented error estimates and mesh adaptation for nonlinear partial differential equations. The setting of the adjoint problem and the resulting estimates are not based on a differentiation of the primal problem but on a suitable linearization which guarantees the adjoint consistency of the numerical scheme. Furthermore, we develop an efficient adaptive algorithm which balances the errors arising from the discretization and the use of nonlinear as well as linear iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.

Goal-Oriented Anisotropic hp-Adaptive Discontinuous Galerkin Method for the Euler Equations

We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin (DG) method with focus on the goal-oriented error estimates and adaptivity. We analyse the adjoint consistency of the DG scheme where the adjoint problem is not formulated by the differentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms. Furthermore, we present the goal-oriented anisotropic hp-mesh adaptation method for the Euler equations. The theoretical results are supported by numerical experiments.

Goal‐oriented adaptivity for linear elastic micromorphic continua based on primal and adjoint consistency analysis

We described an extension of the adaptive finite element (FE) method to linear elastic micromorphic continua using goal-oriented error estimate. A novel patch-based recovery technique for enhanced solutions was developed. The adaptive approach was established on the basis of primal and adjoint consistency analysis, theoretically ensuring an optimal convergence order. We recently have noticed the following two minor errors. For an adjoint consistency analysis, the FE solution is replaced by the exact solution , such that the marked jump terms vanish. The obtained result for adjoint consistency of this paper remains valid.

Analysis and development of adjoint-based h-adaptive direct discontinuous Galerkin method for the compressible Navier–Stokes equations

In this paper, an adjoint-based high-order h-adaptive direct discontinuous Galerkin method is developed and analyzed for the two dimensional steady state compressible Navier–Stokes equations. Particular emphasis is devoted to the analysis of the adjoint consistency for three different direct discontinuous Galerkin discretizations: including the original direct discontinuous Galerkin method (DDG), the direct discontinuous Galerkin method with interface correction (DDG(IC)) and the symmetric direct discontinuous Galerkin method (SDDG). Theoretical analysis shows the extra interface correction term adopted in the DDG(IC) method and the SDDG method plays a key role in preserving the adjoint consistency. To be specific, for the model problem considered in this work, we prove that the original DDG method is not adjoint consistent, while the DDG(IC) method and the SDDG method can be adjoint consistent with appropriate treatment of boundary conditions and correct modifications towards the underlying output functionals. The performance of those three DDG methods is carefully investigated and evaluated through typical test cases. Based on the theoretical analysis, an adjoint-based h-adaptive DDG(IC) method is further developed and evaluated, numerical experiment shows its potential in the applications of adjoint-based adaptation for simulating compressible flows.

Arbitrary High-Order Explicit Hybridizable Discontinuous Galerkin Methods for the Acoustic Wave Equation

We propose a new formulation of explicit time integration for the hybridizable discontinuous Galerkin (HDG) method in the context of the acoustic wave equation based on the arbitrary derivative approach. The method is of arbitrary high order in space and time without restrictions such as the Butcher barrier for Runge–Kutta methods. To maintain the superconvergence property characteristic for HDG spatial discretizations, a special reconstruction step is developed, which is complemented by an adjoint consistency analysis. For a given time step size, this new method is twice as fast compared to a low-storage Runge–Kutta scheme of order four with five stages at polynomial degrees between two and four. Several numerical examples are performed to demonstrate the convergence properties, reveal dispersion and dissipation errors, and show solution behavior in the presence of material discontinuities. Also, we present the combination of local time stepping with h-adaptivity on three-dimensional meshes with curved elements.

Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

This work focuses on multidimensional summation-by-parts (SBP) discretizations of linear elliptic operators with variable coefficients. We consider a general SBP discretization with dense simultaneous approximation terms (SATs), which serve as interior penalties to enforce boundary conditions and inter-element coupling in a weak sense. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties. Based on these conditions, we generalize the modified scheme of Bassi and Rebay (BR2) and the symmetric interior penalty Galerkin (SIPG) method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.

An adjoint-based grid adaptive discontinuous Galerkin method

A grid adaptation method has been developed for discontinuous Galerkin discretizations of two-dimensional Euler equations. The element contribution to the global error in interested output variable is used as an indicator to drive mesh adaptation strategies. In the error indicator construction process, the adjoint variables are obtained by solving the adjoint equations with the GMRES algorithm. The solutions of the primal and adjoint problems in the enriched space of order p +1 are approximated by taking a small number of block-diagonal Jacobi iterations on the projected solutions of order p . In order to enforce the adjoint consistency, the numerical flux and output functions defined on the wall are constructed to depend only on the exterior states. The drag coefficient of NACA0012 airfoil with different orders of accuracy is used to evaluate the reliability of the output error estimation method. The subsonic and transonic inviscid flows around NACA 0012 airfoil, and the Mach 6 hypersonic flow over a half cylinder are simulated using the developed grid adaptation method. The computation results demonstrate that the adjoint-based adaptive method is very effective in reducing discretiztion errors of the output functions such as drag coefficient. The results for the subsonic flow of NACA 0012 shows that only 17% of degrees of freedom employed by the uniform refinement is required to achieve equivalent output accuracy by the adjoint-based adaptive approach.

Goal‐oriented adaptivity for linear elastic micromorphic continua based on primal and adjoint consistency analysis

SummaryMicroscopic considerations are drawing increasing attention for modern simulation techniques. Micromorphic continuum theories, considering micro degrees of freedom, are usually adopted for simulation of localization effects like shear bands. The increased number of degrees of freedom clearly motivates an application of adaptive methods. In this work, the adaptive FEM is tailored for micromorphic elasticity. The proposed adaptive procedure is driven by a goal‐oriented a posteriori error estimator based on duality techniques. For efficient computation of the dual solution, a patch‐based recovery technique is proposed and compared to a reference approach. In order to theoretically ensure optimal convergence order of the proposed adaptive procedure, adjoint consistency of the FE‐discretized solution for the linear elastic micromorphic continua is shown. For illustration, numerical examples are provided. Copyright © 2017 John Wiley & Sons, Ltd.

Generalized adjoint consistent treatment of wall boundary conditions for compressible flows

In this article, we revisit the adjoint consistency analysis of Discontinuous Galerkin discretizations of the compressible Euler and Navier–Stokes equations with application to the Reynolds-averaged Navier–Stokes and k–ω turbulence equations. Here, particular emphasis is laid on the discretization of wall boundary conditions. While previously only one specific combination of discretizations of wall boundary conditions and of aerodynamic force coefficients has been shown to give an adjoint consistent discretization, in this article we generalize this analysis and provide a discretization of the force coefficients for any consistent discretization of wall boundary conditions. Furthermore, we demonstrate that a related evaluation of the cp- and cf-distributions is required. The freedom gained in choosing the discretization of boundary conditions without loosing adjoint consistency is used to devise a new adjoint consistent discretization including numerical fluxes on the wall boundary which is more robust than the adjoint consistent discretization known up to now.While this work is presented in the framework of Discontinuous Galerkin discretizations, the insight gained is also applicable to (and thus valuable for) other discretization schemes. In particular, the discretization of integral quantities, like the drag, lift and moment coefficients, as well as the discretization of local quantities at the wall like surface pressure and skin friction should follow as closely as possible the discretization of the flow equations and boundary conditions at the wall boundary.

Approaches for Adjoint-Based A Posteriori Analysis of Stabilized Finite Element Methods

This study derives a posteriori error estimates for linear functionals of the solution of systems of partial differential equations discretized using stabilized continuous Galerkin methods. We investigate a convection-diffusion equation, the Stokes equations, and incompressible low Reynolds number flow governed by the Navier--Stokes equations. We consider three well-known stabilization methods and show that only one of the three is adjoint consistent and that even this case is contingent upon a proper treatment of the adjoint data. A standard approach for a posteriori error analysis uses the adjoint of the stabilized formulation, which inherits the difficulties induced by the lack of adjoint consistency. We introduce and analyze two alternative approaches. The first is based on the addition of stabilization terms to the adjoint data, while the second is based on a stabilized formulation of the formal adjoint problem. We show that any of the three approaches can be used to derive a fully computable error representation. However, numerical results show that these new alternative approaches result in more accurate error estimates for a variety of problems.

An adjoint consistency analysis for a class of hybrid mixed methods

Hybrid methods represent a classic discretization paradigm for elliptic equations. More recently, hybrid methods have been formulated for convection-diffusion problems, in particular compressible fluid flow. In [25], we have introduced a hybrid mixed method for the compressible Navier-Stokes equations as a combination of a hybridized DG scheme for the convective terms, and an H(div,Ω)-method for the diffusive part. Since hybrid methods are based on Galerkin’s principle, the adjoint of a given hybrid discretization may be used for PDE-constraint optimal control problems, or error estimation, provided that the discretization is adjoint consistent. In the present paper, we extend the adjoint consistency analysis, previously reported for many DG schemes to the more complex hybrid methods. We prove adjoint consistency for a class of Hybrid Mixed schemes, which includes the hybridized DG schemes proposed by [19], as well as our recently proposed method ([25]). Hybrid Mixed discretizations, Hybridized Discontinuous Galerkin discretizations, Adjoint Consistency, compressible Navier-Stokes equations

Adjoint consistency analysis of residual-based variational multiscale methods

We investigate the conditions under which residual-based variational multiscale methods are adjoint, or dual, consistent for model hyperbolic and elliptic partial differential equations. In particular, while many residual-based variational multiscale stabilizations are adjoint consistent for hyperbolic problems and finite-element spaces, only a few are adjoint consistent for elliptic problems.

Constrained C0 Finite Element Methods for Biharmonic Problem

This paper presents some constrained C0 finite element approximation methods for the biharmonic problem, which include the C0 symmetric interior penalty method, the C0 nonsymmetric interior penalty method, and the C0 nonsymmetric superpenalty method. In the finite element spaces, the C1 continuity across the interelement boundaries is obtained weakly by the constrained condition. For the C0 symmetric interior penalty method, the optimal error estimates in the broken H2 norm and in the L2 norm are derived. However, for the C0 nonsymmetric interior penalty method, the error estimate in the broken H2 norm is optimal and the error estimate in the L2 norm is suboptimal because of the lack of adjoint consistency. To obtain the optimal L2 error estimate, the C0 nonsymmetric superpenalty method is introduced and the optimal L2 error estimate is derived.

On the adjoint-consistent formulation of interface conditions in goal-oriented error estimation and adaptivity for fluid–structure interaction

The numerical solution of fluid–structure-interaction problems poses a paradox in that most of the computational resources are consumed by the subsystem of least practical interest, viz., the fluid. Goal-oriented adaptive discretization methods provide a paradigm to bypass this paradox. Based on the solution of a dual problem, the contribution of local residuals to the error in a specific goal functional is estimated, and only the regions that yield a dominant contribution are refined. In the present work, we address a fundamental complication in the application of goal-oriented adaptivity to fluid–structure-interaction problems, namely, that the treatment of the interface conditions has nontrivial consequences for the properties of the dual problem. In the context of a linearized model problem, we consider two equivalent discretizations differing only on the formulation of the interface coupling terms. By means of an adjoint consistency analysis, we show that only one of these discretizations is adjoint consistent. Numerical experiments convey that the two discretizations behave very differently for the dual problem, and that the adjoint-consistent discretization yields more reliable error estimates. Based on the adjoint-consistent discretization, we finally present some h- and hp-adaptive results, confirming that tremendous savings in computational cost can be realized through the use of goal-oriented refinement strategies. The numerical experiments illustrate that the goal-oriented approach effectively equilibrates the error contributions of the fluid and structure subsystems, which is imperative for efficiently resolving the coupled fluid–structure-interaction problem, and which cannot be accomplished by uniform or residual-based refinement strategies.

Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations

This paper is concerned with the adjoint consistency of discontinuous Galerkin (DG) discretizations. Adjoint consistency—in addition to consistency—is the key requirement for DG discretizations to be of optimal order in $L^2$ as well as measured in terms of target functionals. We provide a general framework for analyzing the adjoint consistency of DG discretizations which is also useful for the derivation of adjoint consistent methods. This analysis will be performed for the DG discretizations of the linear advection equation, the interior penalty DG method for elliptic problems, and the DG discretization of the compressible Euler equations. This framework is then used to derive an adjoint consistent DG discretization of the compressible Navier-Stokes equations. Numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions.