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## On adaptive anisotropic mesh optimisation for convection-diffusion problems (2012)

### Citations

3217 | Numerical Optimization
- Nocedal, Wright
- 1999
(Show Context)
Citation Context ...oblem 1 still is a smooth nonlinear inequality constrained optimisation problem. For problems of this class sequential quadratic programming (SQP) is established as a reliable and efficient technique =-=[10, 17]-=-. SQP requires a quadratic model of the (reduced) performance function, which in this work is defined using the BFGS update, thus in the end the gradient information provided with the discrete adjoint... |

585 |
A posteriori error estimation in finite element analysis
- Ainsworth, Oden
- 1997
(Show Context)
Citation Context ...parameter ε > 0. Besides Example 1 we will use the following two examples, for which the exact solution u is unknown. Example 2. : We consider the problem (1) with the L-shape domain Ω = (−1, 1) 2 \( =-=[0, 1]-=-×[−1, 0] ), ΓN = ( {1} × (0, 1) ) ∪ ( (−1, 1) × {1} ), ΓD = Γ \ ΓN, b = (1, 1 2 )T , f = 1, g = 0, and ub = 0. Example 3. [8, Example 6.2]: We consider the problem (1) with Ω = (0, 1) 2 , ΓN = {0} × (... |

521 |
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations
- Brooks, Hughes
- 1982
(Show Context)
Citation Context ...oscillations in the singularly perturbed case. A possible remedy is to add a stabilising term to (3). One of the most popular stabilised methods for convection-dominated problems is the 5SUPG method =-=[2]-=- which combines good stability properties with a high accuracy outside layers. In our case, it can be formulated in the form: Find uh ∈ Wh such that uh = ubh on ΓD and a(uh, vh) + (b · ∇uh, δ b · ∇vh)... |

107 | An Introduction to the Adjoint Approach to design
- GILES, PIERCE
- 2000
(Show Context)
Citation Context ...is whole approach feasible. We treat this by considering the reduced performance function, where the solution operator of (4) is inserted into the performance function. The discrete adjoint technique =-=[6, 16]-=- allows very efficient evaluation of the gradient of this reduced performance function, thus providing everything that is required for an efficient numerical optimisation solver (SQP-type) for this re... |

74 |
Robust Numerical Methods for Singularly Perturbed Differential Equations
- Roos, Stynes, et al.
- 2008
(Show Context)
Citation Context ...on of the problem (1) reads: Find uh ∈ Wh such that uh = ubh on ΓD and a(uh, vh) = (f, vh) + (g, vh)ΓN ∀ vh ∈ Vh := Wh ∩ V . (3) Again, this problem is uniquely solvable. It is well known (cf., e.g., =-=[11]-=-) that the Galerkin discretisation of (1) provides solutions polluted by spurious oscillations in the singularly perturbed case. A possible remedy is to add a stabilising term to (3). One of the most ... |

43 |
Finite element methods for second order differential equations with significant first derivatives
- Griffiths, Mitchell, et al.
- 1976
(Show Context)
Citation Context ...es whether the problem is locally (i.e., within a particular element) convection dominated or diffusion dominated. The formula (5) is a generalisation of a formula derived in the one–dimensional case =-=[3]-=- to obtain a nodally exact solution for constant data and a uniform division of Ω. Let bK be the value of b at the barycentre of K. If bK = 0, the parameter hK is often defined as the diameter of K i... |

39 |
Robust a posteriori error estimates for stationary convection-diffusion equations
- Verfürth
- 2005
(Show Context)
Citation Context ... seem to be preferable for this type of mesh optimisation. Therefore, in the present paper, estimators based on the solution of local problems will be considered. We employ local problems proposed in =-=[18]-=-, however, since the resulting error estimator of [18] was analysed for shape-regular meshes only, we consider also variants of this estimator that differ in the norm used to measure the solutions of ... |

32 |
Numerische Verfahren der nichtlinearen Optimierung
- Spellucci
- 1993
(Show Context)
Citation Context ...oblem 1 still is a smooth nonlinear inequality constrained optimisation problem. For problems of this class sequential quadratic programming (SQP) is established as a reliable and efficient technique =-=[10, 17]-=-. SQP requires a quadratic model of the (reduced) performance function, which in this work is defined using the BFGS update, thus in the end the gradient information provided with the discrete adjoint... |

27 |
Layer-adapted grids for singular perturbation problems. report MATHNM-03-1997
- Roos
- 1997
(Show Context)
Citation Context ...as they typically occur on length scales which are prohibitive to be resolved by standard uniform meshes. As a remedy layer-adapted anisotropic meshes of different types have been developed, see e.g. =-=[12]-=- for an overview. While these meshes are fairly simple to create and very successful in overcoming the problems associated with the layer phenomena, they rely on a priori analysis of the problem, whic... |

22 |
On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: part I – a review’, Comput
- John, Knobloch
- 2007
(Show Context)
Citation Context ...ious that the problem (4) also has a unique solution. The choice of δ strongly influences the quality of the approximate solution, but an optimal way of choosing δ is not known, see the discussion in =-=[7]-=-. In this paper δ will be defined on each triangle K ∈ Th by the constant value hK δ| K = 2 ‖b‖0,∞,K ( coth PeK − 1 PeK ) with PeK = hK ‖b‖ 0,∞,K 2 ε , (5) where hK is a characteristic dimension of K ... |

10 | Stabilized finite element methods with shock capturing for advection–diffusion problems - Knopp, Lube, et al. - 2002 |

9 | Toward anisotropic mesh adaptation based upon sensitivity of a posteriori estimates - Schneider, Jimack - 2005 |

8 |
Superconvergence analysis of the SDFEM for elliptic problems with characteristic layers
- Franz, Linß, et al.
(Show Context)
Citation Context ...uniform mesh of triangles, with 9 by 9 nodes. For all 0 < a ≤ 1/2, 0 < c ≤ 1/4, within each of the sub-domains Ω11, Ω12, Ω21, Ω22, Ω31, Ω32 the nodes are equidistantly spaced in x and y direction. In =-=[5]-=- meshes of this type have been analysed, with the result that the parameters a = aS := min{1/2, 5 ε log(n)}, 2 c = min{1/4, 5√ ε log(n)}, 2 with n = 9 in our case, are suitable for the given setting. ... |

5 | Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problem with characteristic layers - Franz, Linß |

5 |
A posteriori H 1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes
- Kunert
(Show Context)
Citation Context ...timator [1, Section 2.2]) are generally not suitable to exploit the anisotropy of the solution. Extended versions of this approach define a longest and a shortest directional size of an element, e.g. =-=[9]-=-, to allow utilisation of the anisotropy. Unfortunately, the definition of the length scales of the elements in [9] and related works is based on choosing the edge of maximum length, and is thus not c... |

2 | Applications of the Discrete Adjoint Method in Computational Fluid Dynamics
- SCHNEIDER
- 2006
(Show Context)
Citation Context ...plied to the solution of reaction–diffusion problems and the a posteriori error estimation was based on the dual weighted residual method. The aim of the present paper is to apply the general idea of =-=[13]-=- to problem (1) and to adapt the mesh in such a way that an a posteriori estimator of the overall error of the approximate solution is reduced. For this approach to be successful, the a posteriori err... |

1 | On the evaluation of finite element sensitivities to nodal coordinates
- Schneider, Jimack
(Show Context)
Citation Context ...is whole approach feasible. We treat this by considering the reduced performance function, where the solution operator of (4) is inserted into the performance function. The discrete adjoint technique =-=[6, 16]-=- allows very efficient evaluation of the gradient of this reduced performance function, thus providing everything that is required for an efficient numerical optimisation solver (SQP-type) for this re... |