Combining multiresolution and anisotropy Theory, algorithms and open - PowerPoint PPT Presentation

Combining multiresolution and anisotropy Theory, algorithms and open problems Albert Cohen Laboratoire Jacques-Louis Lions Universit´ e Pierre et Marie Curie Paris Joint work with Nira Dyn, Fr´ ed´ eric Hecht and Jean-Marie Mirebeau 22-01-2009

Agenda 1. Multiresolution and adaptivity for PDE’s 2. Optimal anisotropic mesh adaptation 3. A multiresolution approach to anisotropic meshes 4. Numerical illustrations 5. Open problems

Adaptive numerical methods for PDE’s The solution u is discretized on a non-uniform mesh T in which the resolution is locally adapted to its singularities (shocks, boundary layers, sharp gradients... ). Goal: better trade-off between accuracy and CPU/memory space. The adaptive mesh is updated based on the a-posteriori information gained through the computation: ( u 0 , T 0 ) → ( u 1 , T 1 ) → · · · → ( u n , T n ) → ( u n +1 , T n +1 ) → · · · Two typical instances: Steady state problems F ( u ) = 0: the mesh T n is refined according to local error indicators (for example based on residual F ( u n )) and u n → u as n → + ∞ Evolution problems ∂ t u = E ( u ): the numerical solution u n approximates u ( · , n ∆ t ) and the mesh T n is dynamically updated from time step n to n + 1.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n .

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n .

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n .

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n .

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n .

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n .

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n . Evolution problems : T n +1 obtained from T n by refinement and coarsening.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n . Evolution problems : T n +1 obtained from T n by refinement and coarsening.

Multiresolution and adaptivity Adaptive meshes are constrained to be designed by a proper selection of the local resolution within a nested hierarchy of meshes. Steady state problems: T n +1 is a refinement of T n . Evolution problems : T n +1 obtained from T n by refinement and coarsening.

Advantages of multiresolution adaptive methods - Steady states problems: the finite element spaces V n associated to T n are nested, i.e. V n ⊂ V n +1 . Key property for convergence analysis (Doerfler, Morin-Nochetto- Siebert, Binev-DeVore-Dahmen, Stevenson).

Advantages of multiresolution adaptive methods - Steady states problems: the finite element spaces V n associated to T n are nested, i.e. V n ⊂ V n +1 . Key property for convergence analysis (Doerfler, Morin-Nochetto- Siebert, Binev-DeVore-Dahmen, Stevenson). - Evolution problems: mutliresolution framework based on (fine to coarse) re- striction and (coarse to fine) prediction operators (Harten) allow fast adaptive computation of the solution with mass conservation.

Advantages of multiresolution adaptive methods - Steady states problems: the finite element spaces V n associated to T n are nested, i.e. V n ⊂ V n +1 . Key property for convergence analysis (Doerfler, Morin-Nochetto- Siebert, Binev-DeVore-Dahmen, Stevenson). - Evolution problems: mutliresolution framework based on (fine to coarse) re- striction and (coarse to fine) prediction operators (Harten) allow fast adaptive computation of the solution with mass conservation. - Possibility of computing wavelets coefficients allowing to perform local smooth- ness analysis of the numerical solution and adaptive coarsening by thresholding.

Advantages of multiresolution adaptive methods - Steady states problems: the finite element spaces V n associated to T n are nested, i.e. V n ⊂ V n +1 . Key property for convergence analysis (Doerfler, Morin-Nochetto- Siebert, Binev-DeVore-Dahmen, Stevenson). - Evolution problems: mutliresolution framework based on (fine to coarse) re- striction and (coarse to fine) prediction operators (Harten) allow fast adaptive computation of the solution with mass conservation. - Possibility of computing wavelets coefficients allowing to perform local smooth- ness analysis of the numerical solution and adaptive coarsening by thresholding. - Fast encoding: an adaptive mesh can be seen as a finite tree selected within an infinite binary decision tree. A tree with N leaves can be encoded in 2 N bits.

Advantages of multiresolution adaptive methods - Steady states problems: the finite element spaces V n associated to T n are nested, i.e. V n ⊂ V n +1 . Key property for convergence analysis (Doerfler, Morin-Nochetto- Siebert, Binev-DeVore-Dahmen, Stevenson). - Evolution problems: mutliresolution framework based on (fine to coarse) re- striction and (coarse to fine) prediction operators (Harten) allow fast adaptive computation of the solution with mass conservation. - Possibility of computing wavelets coefficients allowing to perform local smooth- ness analysis of the numerical solution and adaptive coarsening by thresholding. - Fast encoding: an adaptive mesh can be seen as a finite tree selected within an infinite binary decision tree. A tree with N leaves can be encoded in 2 N bits. Existing multiresolution approaches mainly based on isotropic refinement while optimally adapted meshes are often anisotropic.

Optimally adapted triangulations Goal: given a function f and N > 0, build triangulation T N with N triangles, which minimizes the L p -distance between f and the P 1 (piecewise linear) finite element space for T N . T N should exhibit anisotropic refinement along the shocks and sharp gradients.

Optimally adapted triangulations Goal: given a function f and N > 0, build triangulation T N with N triangles, which minimizes the L p -distance between f and the P 1 (piecewise linear) finite element space for T N . T N should exhibit anisotropic refinement along the shocks and sharp gradients. Finding the exactly optimal triangulation T N is NP-hard. More reasonable goal: build T N such that the approximation error behaves similar as with an optimal triangulation when N → + ∞ .

Approximation error for P 1 finite elements When ( T h ) h> 0 is a family of quasi-uniform triangulations and V h the associated P 1 finite element space, one has for f smooth enough f h ∈ V h � f − f h � L p ≤ Ch 2 � d 2 f � L p . inf In terms of number of triangles N = #( T h ) ∼ h − 2 , the approximation error is thus controlled by CN − 1 � d 2 f � L p Question: how can we improve this estimate when using a triangulation adapted to f ?

Approximation by adaptive isotropic finite elements Start from a basic estimate for the local error on the triangle T e T ( f ) p := inf π ∈ Π 1 � f − π � L p ( T ) or � f − I T f � L p ( T ) ou � f − P T f � L p ( T ) , with I T the interpolant and P T the L 2 ( T )-orthogonal projector: e T ( f ) p ≤ Ch 2 T � d 2 f � L p ( T ) , with h T := diam( T ). For isotropic triangles: h 2 T ∼ | T | therefore e T ( f ) p ≤ C | T |� d 2 f � L p ( T ) ,