Anisotropic meshes for PDEs: a posteriori error analysis and mesh adaptivity

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ICAM 2013, Heraklion, 17 September 2013 Anisotropic meshes for PDEs: a posteriori error analysis and mesh adaptivity Simona Perotto MOX-Modeling and Scientific Computing

1 ICAM 2013, Heraklion, 17 September 2013 Anisotropic meshes for PDEs: a posteriori error analysis and mesh adaptivity Simona Perotto MOX-Modeling and Scientific Computing Department of Mathematics, Politecnico di Milano joint work with: L. Formaggia, S. Micheletti (MOX- Politecnico di Milano, Italy), P.E. Farrell (Imperial College, London)

properly chosing the size, the shape and the orientation of the mesh elements

2 Isotropy vs anisotropy L. Dedè, S. Micheletti, S. P. (2008) reduce the number of the degrees of freedom for a fixed solution accuracy, by properly chosing the size, the shape and the orientation of the mesh elements advective-diffusive problem + homogeneous Dirichlet b.c. t.c. exact solution u( x, y) x 50 4 x 50 y y 1 e 1 e e 3 boundary layers

properly chosing the size, the shape and the

3 Isotropy vs anisotropy L. Dedè, S. Micheletti, S. P. (2008) reduce the number of the degrees of freedom for a fixed solution accuracy, by properly chosing the size, the shape and the orientation of the mesh elements A 500 elements err = 3.31e-01 CPU time 7. 86s I 3200 elements err = 3.37e-01 CPU time 108s control of the energy longest edges aligned with the boundary layers

properly chosing the size, the shape and the orientation of the mesh elements A 2500 elements err

4 Isotropy vs anisotropy L. Dedè, S. Micheletti, S. P. (2008) maximize the accuracy of the numerical solution for a fixed number of mesh elements, by properly chosing the size, the shape and the orientation of the mesh elements A 2500 elements err = 1.383e-01 CPU time 83. 4s I 2500 elements err = 3.929e-01 CPU time 68. 8s control of the energy longest edges aligned with the boundary layers

5 Isotropy vs anisotropy L. Dedè, S. Micheletti, S. P. (2008) maximize the accuracy of the numerical solution for a fixed number of mesh elements, by properly chosing the size, the shape and the orientation of the mesh elements

6 ICAM 2013, Heraklion, 17 September 2013 Anisotropic meshes for PDEs: a posteriori error analysis and mesh adaptivity The anisotropic setting Simona Perotto Anisotropic estimates: a priori and a posteriori (goal-oriented) estimates Recent developments: anisotropic recovery-based analysis

7 The Anisotropic Setting [L. Formaggia, S.P. (2001)] source of the anisotropic information reference isotropic element 2D ( polar decomposition ) s.p.d. orthogonal

8 First step towards an anisotropic mesh adaptation anisotropic interpolation error estimates Lemma 1: A Let... Then, the following estimate holds: I Hessian matrix the information lumped in the diameter and in the seminorm is split along the anisotropic directions we recover the isotropic estimates when L. Formaggia, S.P. (2001, 2003)

.. Then, the following estimate holds: I Hessian matrix first example:

9 First step towards an anisotropic mesh adaptation anisotropic interpolation error estimates Lemma 1: A Let... Then, the following estimate holds: I Hessian matrix first example: a priori anisotropic mesh adaptation we recover the isotropic estimates when L. Formaggia, S.P. (2001, 2003)

10 Anisotropic a posteriori error analysis L. Formaggia, S.P. (2003) generalization of the goal-oriented a posteriori analysis to an anisotropic context. control of physically meaningful quantities R. Becker, R. Rannacher (1996) T.J. Oden, S. Prudhomme (2001) M. Giles, E. Süli (2002) The goal quantities are mathematically represented by suitable (linear and nonlinear) functionals. dual problem ( lift ) ( drag ) ( convective flux ) (diffusive flux ) ( total surface tension ) ( torsion moment )

Anisotropic a posteriori error analysis control, to

via, instead of a suitable norm of the discretization

necessarily follow the solution J(u) goal-oriented

the potentialities of the two approaches to control

11 Anisotropic a posteriori error analysis control, to within a prescribed tolerance, of the exact value via, instead of a suitable norm of the discretization error grids adapted according to J which do not necessarily follow the solution J(u) goal-oriented analysis anisotropic interpolation estimates: u merge the potentialities of the two approaches to control goal quantities with reduced computational costs and/or high accuracy

12 Anisotropic goal-oriented a posteriori error analysis : an example model problem a.e. in, + ellipticity condition,. weak form (WF) :? : FE approximation (FE) :? : dual problem (DP) :? : adjoint form

13 Proposition : Let be the solution to (WF), the solution to (FE) and the solution to (DP), s.t.. Then, it holds: residuals weights boundary residual internal residual jump of the conormal derivative the anisotropic information is essentially lumped in the weights analysis fully transversal to the PDEs theory

14 Extension to more complex PDEs advection-dominated problems Stokes equations nonlinear problems unsteady problems

15 How to get the adapted mesh? a metric-based adaptive procedure predictive procedure: metric is the actual unknown CRITERION : minimize the number of the degrees of freedom for a fixed solution accuracy estimator metric adapted mesh At each step of the adaptive process: the actual mesh ; evaluation of the error estimator local optimization problems the predicted (piecewise constant) metric computed on ; the updated mesh. BAMG + stopping criterion

16 More recently anisotropic recovery-based error estimators

17 Recovery-based error estimators recovered gradient Zienckiewicz-Zhu error estimator [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)]

18 Recovery-based error estimators Zienckiewicz-Zhu error estimator recovered gradient (ad-hoc averagings/projections of the actual discrete gradient over suitable element or node patches) [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)]

19 Recovery-based error estimators Zienckiewicz-Zhu error estimator recovered gradient (ad-hoc averagings/projections of the actual discrete gradient over suitable element or node patches) [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)] not confined to a specific problem independent of the FE formulation cheap to compute easy to implement work pretty well in practice [e.g., R. Rodríguez (1994), M. Krízek, P. Neittaanmäki (1984,1987), C. Carstensen (2001,2004), A. Naga, Z. Zhang (2004), N. Yan, A. Zhou (2001), X.D. Li, N.-E. Wiberg (1994), C.L. Bottasso, G. Maisano, S. Micheletti, S.P. (2006), J. Xu, Z. Zhang (2004), etc.]

20 Recovery-based error estimators Zienckiewicz-Zhu error estimator recovered gradient (ad-hoc averagings/projections of the actual discrete gradient over suitable element or node patches) [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)] not confined to a specific problem independent of the FE formulation cheap to compute easy to implement work pretty well in practice to combine these good properties with the richness of information typical of an anisotropic error analysis

21 Taking inspiration from [L. Formaggia, S.P. (2001)] the information lumped in the diameter and in the seminorm is split along the anisotropic directions

22 Taking inspiration from [L. Formaggia, S.P. (2001)] the anisotropic counterpart of the H1-seminorm is represented via the matrix

23 Taking inspiration from recovered gradient Zienckiewicz-Zhu error estimator [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)] the anisotropic counterpart of the H1-seminorm is represented via the matrix

24 An anisotropic recovery-based error estimator Zienckiewicz-Zhu error estimator recovered gradient [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)] recovered-based AD-HOC recovered gradient [S. Micheletti, S.P. (2009)]

25 An anisotropic recovery-based error estimator Zienckiewicz-Zhu error estimator recovered gradient [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)] novelty: anisotropic projection AD-HOC recovered gradient [S. Micheletti, S.P. (2009)]

26 An anisotropic recovery-based error estimator Zienckiewicz-Zhu error estimator recovered gradient [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)] scaling factor to recover the isotropic case AD-HOC recovered gradient [S. Micheletti, S.P. (2009)]

An anisotropic recovery-based error estimator Zienckiewicz-Zhu error estimator recovered gradient [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)] K AD-HOC approach successfully extended to a 3D setting recovered gradient [P.

27 An anisotropic recovery-based error estimator Zienckiewicz-Zhu error estimator recovered gradient [O.C. Zienkiewicz, J.Z. Zhu (1987,1992)] K AD-HOC approach successfully extended to a 3D setting recovered gradient [P.E. Farrell, S. Micheletti, S.P. (2010,2011)] [S. Micheletti, S.P. (2009)]

28 Numerical assessment: laminar flow past a cylinder Re m u HD on u 0.2m 2.5m speed of the flow HD on u HD on u HN on u HD on p incompressible Navier-Stokes equations stabilized P1-P1 [M. Schäfer, S. Turek, F. Durst, E. Krause, R. Rannacher (1996)]

29 Numerical assessment: laminar flow past a cylinder Re 0.41m 20 u HD on u 0.2m 2.5m speed of the flow HD on u HD on u HN on u HD on p incompressible Navier-Stokes equations stabilized P1-P nodes isotropic anisotropic P.E. Farrell, S. Micheletti, S.P. (2011)

30 Space-time adaptation G. Porta, S.P., F. Ballio (2012) total water depth unsteady shallow water equations

31 Space-time adaptation G. Porta, S.P., F. Ballio (2012) total water depth unsteady shallow water equations

32 Space-time adaptation G. Porta, S.P., F. Ballio (2012) 0.271s ~ 400 Unsteady shallow water equations: space-time adaptation localized overdiffusion reference solution on a grid with elements and time step = 0.002s

33 Space-time adaptation G. Porta, S.P., F. Ballio (2012) 0.178s ~ 400 localized overdiffusion

Space-time adaptation G. Porta, S.P., F. Ballio (2012) ~ 4000 0.

34 Space-time adaptation G. Porta, S.P., F. Ballio (2012) ~ s Unsteady shallow water equations: space-time adaptation GOOD AGREEMENT slight dispersion

35 Generalization to a goal-oriented setting S. Micheletti, S.P. (2013) H1 seminorm mean value energy norm

36 Generalization to a goal-oriented setting H1 seminorm S. Micheletti, S.P. (2013) mean value order = -1/2 order = -1 H1 seminorm mean value energy norm

Some remarks on the stability coefficients and bubble stabilization of FEM on anisotropic meshes

Some remarks on the stability coefficients and bubble stabilization of FEM on anisotropic meshes Stefano Micheletti Simona Perotto Marco Picasso Abstract In this paper we re-address the anisotropic recipe