-
-We consider the a posteriori error analysis
-and adaptive mesh design for discontinuous Galerkin finite
-element approximations to systems of nonlinear
-hyperbolic conservation laws.
-In particular, we discuss the question of error estimation for general
-linear and nonlinear functionals of the solution; typical examples
-include the
-outflow flux, local average and pointwise value, as well as the lift
-and drag coefficients of a body immersed in an inviscid fluid.
-By employing a duality argument, we derive so-called weighted or Type I
-a posteriori error bounds; in these error estimates
-the element--residuals are multiplied by local weights involving
-the solution of a certain dual problem. Based on these a posteriori
-bounds, we design and implement the corresponding adaptive algorithm
-to ensure efficient and reliable control of the error in the
-computed functional. The theoretical results are illustrated by a series of
-numerical experiments. In particular, we demonstrate the superiority
-of the proposed approach over standard mesh refinement algorithms which employ
-ad hoc error indicators.
-
-
-
-
-In this paper a recently developed approach
- for the design of
- adaptive discontinuous Galerkin finite element approximations
- is applied to physically relevant problems arising in inviscid compressible
- fluid flows governed by the
- Euler equations of gas dynamics. In particular, we employ so--called
- weighted or Type I a posteriori error bounds to drive adaptive
- finite element algorithms for the estimation of
- the error measured in terms of general linear and nonlinear target
- functionals of the solution; typical examples considered here include
- the point evaluation of a component of the solution vector, and the
- drag and lift coefficients of a body immersed in an inviscid fluid.
- 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 reliable and
- efficient error estimation. Indeed, the superiority of the proposed
- approach over standard mesh refinement algorithms which employ
- ad hoc error indicators will be illustrated by a series of
- numerical experiments; here, we consider
- transonic flow through a nozzle, as well as subsonic, transonic and
- supersonic flows around different airfoil geometries.
-
-
-
-
-@InProceedings{HH01a,
- author = {Paul Houston and Ralf Hartmann},
- title = {Goal--Oriented A Posteriori Error Estimation for Compressible Fluid Flows},
- booktitle = {Numerical Mathematics and Advanced Applications},
- pages = {775--784},
- year = 2003,
- editor = {F. Brezzi and A. Buffa and S. Corsaro and A. Murli},
- publisher = {Springer}
-}
-
-
-
-
-We consider so-called `goal-oriented' a posteriori error
-estimation for discontinuous Galerkin finite element approximations to
-the compressible Euler equations of gas dynamics. By employing a
-hyperbolic duality argument, we derive weighted, or Type I, a
-posteriori error estimates which bound the error measured in
-terms of certain target functionals of real or physical interest. The
-practical advantages of this general approach are illustrated by a
-series of numerical experiments.
-
-
-
-@PhdThesis{Har02,
- author = {Ralf Hartmann},
- title = {Adaptive Finite Element Methods for the
- Compressible Euler Equations},
- school = {University of Heidelberg},
- year = 2002
-}
-
-
-
-
-
-In this thesis we introduce a discontinuous Galerkin method for the
-numerical solution of hyperbolic conversation laws, as for example the
-compressible Euler equations of gas dynamics. Based on this finite
-element method, we develop an adaptive algorithm for the efficient
-computation of physically relevant quantities of the solution. This
-includes a posteriori error estimation of the error in the
-computed quantity as well as adaptive mesh design specifically
-tailored to the efficient computation of this quantity. We illustrate
-this approach by several different hyperbolic problems in combination
-with various different target quantities, including the efficient
-computation of drag and lift coefficients of airfoils immersed in
-inviscid compressible gas flows.
-
-
-In particular, this work includes following issues
-
-
- Discretisation:
-
-
Streamline diffusion and discontinuous Galerkin method
- for a scalar hyperbolic problem: comparison with respect to
- accuracy and conservation properties
-
Discontinuous Galerkin method for the compressible Euler equations
-
Use of shock-capturing
-
Higher order boundary approximation at reflective boundaries
-
Solution of the nonlinear and the linear problems
-
Mesh generation for airfoil computations
-
-
-
-
- A posteriori error estimation and adaptivity:
-
-
Derivation (by duality argument) of error estimates with
- respect to arbitrary target functionals
-
-
Question of well-posedness of the dual problem
-
-
Approximation of the error representation by numerical
- approximation of the dual solution
-
-
Adaptive algorithm tailored to the efficient computation
- of the quantity of interest
-
-
-
-
Numerical Examples illustrating the performance of the
- error estimation and the adaptive grid refinement for a wide range
- of hyperbolic problems including
-
-
the linear advection equation
-
the 1D inviscid Burgers equation
-
the Buckley-Leverett equation
-
the 1D compressible Euler equations
-
and several examples for the 2D compressible Euler equations.
-
- and for a variety of target functionals (quantities) including
-
-
drag and lift coefficients of airfoils
-
pressure point values
-
weighted boundary integrals
-
-
-
-
- Appendices
-
-
Full description of exact solutions to hyperbolic
- problems treated in numerical examples
-
-
Implementational details of finite elements with curved
- boundaries
-
-
- In many applications the quantities of interest are a series of
- target functionals of the solution to the governing system of
- partial differential equations rather than the solution itself. For
- example, in the field of aerodynamics, examples include the drag and
- lift coefficients of an airfoil immersed into a fluid, the pressure
- difference between the leading and trailing edges of the airfoil and
- point evaluations of the density or pressure on the profile of the
- airfoil. While traditionally these quantities are measured in wind
- tunnel experiments, nowadays these experiments are increasingly
- replaced by numerical simulations aiming to predict these quantities
- to a high level of accuracy.
-
- In a series of previous articles, we have developed the theory of
- goal--oriented a posteriori error estimation for
- discontinuous Galerkin methods applied to inviscid compressible
- fluid flows. On the basis of Type I a posteriori bounds we
- considered the design of adaptive finite element algorithms that are
- capable of generating optimal meshes specifically tailored to the
- efficient computation of a single target functional of
- practical interest. The purpose of the current article is to extend
- this earlier work to the case when several target
- functionals of the solution need to be simultaneously approximated
- to a given level of accuracy.
-