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On adaptive timestepping for weakly instationary solutions of hyperbolic conservation laws via adjoint error control
 Comm. Numer. Meth. Eng
, 2008
"... Abstract. We study a recent timestep adaptation technique for hyperbolic conservation laws. The key tool is a spacetime splitting of adjoint error representations for target functionals due to Süli[19] and Hartmann[13]. It provides an efficient choice of timesteps for implicit computations of weakl ..."
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Abstract. We study a recent timestep adaptation technique for hyperbolic conservation laws. The key tool is a spacetime splitting of adjoint error representations for target functionals due to Süli[19] and Hartmann[13]. It provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow, and become small when a perturbation enters the flow field. Besides using adjoint techniques which are already wellestablished, we also add a new ingredient which simplifies the computation of the dual problem. Due to Galerkin orthogonality, the dual solution ϕ does not enter the error representation as such. Instead, the relevant term is the difference of the dual solution and its projection to the finite element space, ϕ −ϕh. We can show that it is therefore sufficient to compute the spatial gradient of the dual solution, w = ∇ϕ. This gradient satisfies a conservation law instead of a transport equation, and it can therefore be computed with the same algorithm as the forward problem, and in the same finite element space. We demonstrate the capabilities of the approach for a weakly instationary test problem for scalar conservation laws. Contents
Finite element algorithms for transportdiffusion problems: stability, adaptivity, tractability
 Invited Lecture at the International Congress of Mathematicians
, 2006
"... Abstract. Partial differential equations with nonnegative characteristic form arise in numerous mathematical models of physical phenomena: stochastic analysis, in particular, is a fertile source of equations of this kind. We survey recent developments concerning the finite element approximation of t ..."
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Abstract. Partial differential equations with nonnegative characteristic form arise in numerous mathematical models of physical phenomena: stochastic analysis, in particular, is a fertile source of equations of this kind. We survey recent developments concerning the finite element approximation of these equations, focusing on three relevant aspects: (a) stability and stabilisation; (b) hpadaptive algorithms driven by residualbased a posteriori error bounds, capable of automatic variation of the granularity h of the finite element partition and of the local polynomial degree p; (c) complexityreduction for highdimensional transportdiffusion problems by stabilised sparse finite element methods.
ADAPTIVE TIMESTEP CONTROL FOR INSTATIONARY SOLUTIONS OF THE EULER EQUATIONS
"... In this paper we continue our work on adaptive timestep control for weakly instationary problems [29]. The core of the method is a spacetime splitting of adjoint error representations for target functionals due to Süli [31] and Hartmann [18]. The main new ingredients are (i) the extension from sca ..."
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Cited by 1 (1 self)
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In this paper we continue our work on adaptive timestep control for weakly instationary problems [29]. The core of the method is a spacetime splitting of adjoint error representations for target functionals due to Süli [31] and Hartmann [18]. The main new ingredients are (i) the extension from scalar, 1D, conservation laws to the 2D Euler equations of gas dynamics, (ii) the derivation of boundary conditions for a new formulation of the adjoint problem and (iii) the coupling of the adaptive timestepping with spatial adaptation. For the spatial adaptation, we use a multiscalebased strategy developed by Müller [24], and we combine this with an implicit time discretization. The combined spacetime adaptive method provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow, and becomes small when a perturbation enters the flow field. The efficiency of the solver is investigated by means of an unsteady inviscid 2D flow over a bump.
On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows
, 2002
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BY
, 2005
"... This is to certify that I have examined this copy of a doctoral thesis by ..."
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This is to certify that I have examined this copy of a doctoral thesis by
Computing and Information SYMMETRIC INTERIOR PENALTY DG METHODS FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS II: GOAL–ORIENTED A POSTERIORI ERROR ESTIMATION
"... Abstract. In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier– Stokes equations. In particular, we consider the a posteriori error ana ..."
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Abstract. In this article we consider the application of the generalization of the symmetric version of the interior penalty discontinuous Galerkin finite element method to the numerical approximation of the compressible Navier– Stokes equations. In particular, we consider the a posteriori error analysis and adaptive mesh design for the underlying discretization method. Indeed, by employing a duality argument (weighted) Type I a posteriori bounds are derived for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing efficient error estimation. Numerical experiments demonstrating the performance of the proposed approach will be presented. Key Words. Discontinuous Galerkin methods, a posteriori error estimation, adaptivity, compressible Navier–Stokes equations 1.
control for the HamiltonJacobi equations. Part I: The onedimensional steady state case
, 2004
"... In this paper, we introduce a new adaptive method for finding approximations for HamiltonJacobi equations whose L ∞distance to the viscosity solution is no bigger than a prescribed tolerance. This is done on the simple setting of a onedimensional model problem with periodic boundary conditions. W ..."
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In this paper, we introduce a new adaptive method for finding approximations for HamiltonJacobi equations whose L ∞distance to the viscosity solution is no bigger than a prescribed tolerance. This is done on the simple setting of a onedimensional model problem with periodic boundary conditions. We consider this to be a stepping stone towards the more challenging goal of contructing such methods for general HamiltonJacobi equations. The method proceeds as follows. On any given grid, the approximate solution is computed by using a well known monotone scheme; then, the quality of the approximation is tested by using an approximate a posteriori error estimate. If the error is bigger than the prescribed tolerance, a new grid is computed by solving a differential equation whose devising is the main contribution of the paper. A thorough numerical study of the method is performed which shows that rigorous error control is achieved, even though only an approximate a posteriori error estimate is used; the method is thus reliable. Furthermore, the numerical study also shows that the method is efficient and that it has an optimal computational
DISCONTINUOUS GALERKIN METHODS FOR ELLIPTIC PROBLEMS
, 2004
"... Abstract. In this paper, we develop the a posteriori error estimation of hpversion interior penalty discontinuous Galerkin discretizations of elliptic boundaryvalue problems. Computable upper and lower bounds on the error measured in terms of a natural (meshdependent) energy norm are derived. The ..."
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Abstract. In this paper, we develop the a posteriori error estimation of hpversion interior penalty discontinuous Galerkin discretizations of elliptic boundaryvalue problems. Computable upper and lower bounds on the error measured in terms of a natural (meshdependent) energy norm are derived. The bounds are explicit in the local mesh sizes and approximation orders. A series of numerical experiments illustrate the performance of the proposed estimators within an automatic hpadaptive refinement procedure. Key words. Discontinuous Galerkin methods, a posteriori error estimation, hpadaptivity, elliptic problems. AMS subject classifications. 65N30, 65N35, 65N50
Bounds on Quantities of Physical Interest
"... In many computational simulations there are identifiable quantities of physical interest which are weighted integrals of the solution of a boundary or initial value problem. Examples include lift and drag in aerodynamics and well productions in reservoir simulations. Methods that can bound these qua ..."
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In many computational simulations there are identifiable quantities of physical interest which are weighted integrals of the solution of a boundary or initial value problem. Examples include lift and drag in aerodynamics and well productions in reservoir simulations. Methods that can bound these quantities sharply are therefore of considerable practical importance. Although existing theory can sometimes be used to construct theoretical bounds on these quantities, for example those governed by a selfadjoint operator, it has not hitherto been exploited in a practical context. In this thesis, novel applications of these bounds have been devised and implemented for a number of model problems. In particular, the upscaling problem experienced in the oil industry, which is governed by the steady state diffusion equation, is addressed by finding bounds on the well outflow. Extensions to the theory for problems involving nonselfadjoint operators, which enable computable bounds to be determined for quantities of physical interest, are developed using two approaches. Firstly, semidiscrete approximations are considered for the timedependent diffusion equations and the advectiondiffusion equation, although this
Adaptive Finite Element Methods for Local Volatility European Option Pricing
, 2002
"... CERMICS, Ecole nationale des ponts et chaussées ..."