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119
Grid adaptation for functional outputs: application to twodimensional inviscid flows
 J. Comput. Phys
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A triangular cutcell adaptive method for highorder discretizations of the compressible Navier–Stokes equations
, 2007
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An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method
, 2005
"... Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG103035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higherorder DG. In particular, Krzysztof Fidkowsk ..."
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Cited by 29 (0 self)
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Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG103035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higherorder DG. In particular, Krzysztof Fidkowski and Todd Oliver are to be acknowledged for their contributions towards the development of the flow solvers and also for providing some of the grids for the test cases demonstrated. Finally, thanks must go to thesis committee members Professors Peraire and Willcox as well as thesis readers Dr. Natalia Alexandrov and Dr. Steven Allmaras for the time they put into reading the thesis and providing the valuable feedbacks. 3 46 Adjoint approach to shape sensitivity 117 6.1 Introduction...............................
Adjoint consistency analysis of discontinuous Galerkin discretizations
 SIAM J. Numer. Anal
"... Abstract. 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 L2 as well as measured in terms of target functionals. We provide a ge ..."
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Cited by 27 (5 self)
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Abstract. 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 L2 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.
Symmetric interior penalty DG methods for the compressible Navier{Stokes equations II: Goal{oriented a posteriori error estimation
 In preparation
"... Abstract. In this article we consider the development of discontinuous Galerkin nite element methods for the numerical approximation of the compressible Navier{Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the ..."
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Abstract. In this article we consider the development of discontinuous Galerkin nite element methods for the numerical approximation of the compressible Navier{Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear selfadjoint second{order elliptic partial dierential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton{GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher{order polynomials are presented. Key words. Finite element methods, discontinuous Galerkin methods, compressible Navier{ Stokes equations 1.
Algebraic flux correction II. Compressible Euler equations
 FluxCorrected Transport: Principles, Algorithms, and Applications
, 2005
"... Summary. Algebraic flux correction schemes of TVD and FCT type are extended to systems of hyperbolic conservation laws. The group finite element formulation is employed for the treatment of the compressible Euler equations. An efficient algorithm is proposed for the edgebyedge matrix assembly. A ..."
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Summary. Algebraic flux correction schemes of TVD and FCT type are extended to systems of hyperbolic conservation laws. The group finite element formulation is employed for the treatment of the compressible Euler equations. An efficient algorithm is proposed for the edgebyedge matrix assembly. A generalization of Roe’s approximate Riemann solver is derived by rendering all offdiagonal matrix blocks positive semidefinite. Another usable loworder method is constructed by adding scalar artificial viscosity proportional to the spectral radius of the cumulative Roe matrix. The limiting of antidiffusive fluxes is performed using a transformation to the characteristic variables or a suitable synchronization of correction factors for the conservative ones. The outer defect correction loop is equipped with a blockdiagonal preconditioner so as to decouple the discretized Euler equations and solve them in a segregated fashion. As an alternative, a strongly coupled solution strategy (global BiCGSTAB method with a blockGaußSeidel preconditioner) is introduced for applications which call for the use of large time steps. Various algorithmic aspects including the implementation of characteristic boundary conditions are addressed. Simulation results are presented for inviscid flows in a wide range of Mach numbers. 1
DISCONTINUOUS GALERKIN FINITE ELEMENT APPROXIMATION OF NONLINEAR SECONDORDER ELLIPTIC AND HYPERBOLIC SYSTEMS
"... Abstract. We develop the convergence analysis of discontinuous Galerkin finite element approximations to symmetric secondorder quasilinear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in R d, subject to mixed Dirichlet–Neumann boun ..."
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Abstract. We develop the convergence analysis of discontinuous Galerkin finite element approximations to symmetric secondorder quasilinear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in R d, subject to mixed Dirichlet–Neumann boundary conditions. Optimalorder asymptotic bounds are derived on the discretization error in each case without requiring the global Lipschitz continuity or uniform monotonicity of the stress tensor. Instead, only local smoothness and a G˚arding inequality are used in the analysis. Key words. Nonlinear elliptic and hyperbolic systems of partial differential equations, discontinuous Galerkin methods, Legendre–Hadamard condition, broken G˚arding inequality 1. Introduction. Secondorder nonlinear elliptic and hyperbolic systems of partial differential equations arise in numerous applications, and a substantial body of research has been devoted to their analytical and computational study. This paper is concerned with the construction and convergence analysis of a class of numerical algorithms — discontinuous Galerkin finite element methods — for the approximate solution of quasilinear elliptic and hyperbolic systems. Nonlinear elasticity is a
A posteriori error analysis for higher order dissipative methods for evolution problems
"... Abstract. We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method and the corresponding implicit RungeKuttaRadau method of arbitrary order for both linear and nonlinear evolution problems. The key ingredient is a novel higher order reconstruction Û of th ..."
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Cited by 15 (3 self)
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Abstract. We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method and the corresponding implicit RungeKuttaRadau method of arbitrary order for both linear and nonlinear evolution problems. The key ingredient is a novel higher order reconstruction Û of the discrete solution U, which restores continuity and leads to the differential equation Û ′ +ΠF(U) = F for a suitable interpolation operator Π. The error analysis hinges on careful energy arguments and the monotonicity of the operator F, in particular its angle bounded structure. We discuss applications to linear PDE such as the convectiondiffusion equation and the wave equation, and nonlinear PDE corresponding to subgradient operators such as the pLaplacian and minimal surfaces, as well as Lipschitz and noncoercive operators. 1.
Shock Capturing with PDEBased Artificial Viscosity for DGFEM: Part I, Formulation
"... Artificial viscosity can be combined with a higherorder discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a nonsmooth artificial viscosity model is employed with an otherwise higherorder approximation, elementtoelement variations in ..."
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Artificial viscosity can be combined with a higherorder discontinuous Galerkin finite element discretization to resolve a shock layer within a single cell. However, when a nonsmooth artificial viscosity model is employed with an otherwise higherorder approximation, elementtoelement variations induce oscillations in state gradients and pollute the downstream flow. To alleviate these difficulties, this work proposes a higherorder, statebased artificial viscosity with an associated governing partial differential equation (PDE). In the governing PDE, a shock indicator acts as a forcing term while gridbased diffusion is added to smooth the resulting artificial viscosity. When applied to heat transfer prediction on unstructured meshes in hypersonic flows, the PDEbased artificial viscosity is less susceptible to errors introduced by grid edges oblique to captured shocks and boundary layers, thereby enabling accurate heat transfer predictions.
Review of OutputBased Error Estimation and Mesh Adaptation in Computational Fluid Dynamics
"... Error estimation and control are critical ingredients for improving the reliability of computational simulations. Adjointbased techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement. This article reviews recent work on th ..."
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Error estimation and control are critical ingredients for improving the reliability of computational simulations. Adjointbased techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement. This article reviews recent work on these techniques for computational fluid dynamics applications in aerospace engineering. The definition of the adjoint as the sensitivity of an output to residual source perturbations is used to derive both the adjoint equation, in fully discrete and variational formulations, and the adjointweighted residual method for error estimation. Assumptions and approximations made in the calculations are discussed. Presentation of the discrete and variational formulations enables a sidebyside comparison of recent work in outputerror estimation using the finite volume method and the finite element method. Techniques for adapting meshes using outputerror indicators are also reviewed. Recent adaptive results from a variety of laminar and Reynoldsaveraged Navier–Stokes applications show the power of outputbased adaptive methods for improving the robustness of computational fluid dynamics computations. However, challenges and areas of additional future research remain, including computable error bounds and robust mesh adaptation mechanics. I.