Backgrounds Mathematical theory of anisotropic FEM Talk from Netwon’s interpolation Anisotropic finite element methods with their applications polynomial From the viewpoint of Girault- Raviart interpolation From the viewpoint of orthogonal Shipeng M AO expansions 2 Anisotropic H elements Degenerate Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, 100190, quadrilateral Beijing, China elements Application to problems with singularities MOMAS, Paris, 04/12/2008 Application to singularly perturbed problems LSEC Shipeng M AO Anisotropic finite element methods with their applications 1/64
Backgrounds Mathematical theory of anisotropic FEM Talk from Netwon’s interpolation polynomial From the viewpoint of Girault- Raviart interpolation From the viewpoint of Backgrounds 1 orthogonal expansions 2 Anisotropic H elements Degenerate quadrilateral elements Application to problems with singularities Application to singularly perturbed problems LSEC Shipeng M AO Anisotropic finite element methods with their applications 2/64
Anisotropic FEM VS Isotropic FEM Backgrounds Mathematical Anisotropic FEM (regularity assumption in Ciarlet [1978] or nondegenerate theory of anisotropic condition in Brenner, Scott [1994]):The length of the longest edge should be FEM Talk from comparable with that of the diameter of the inscribed ball Netwon’s interpolation polynomial From the viewpoint of Girault- Raviart interpolation From the viewpoint of orthogonal expansions 2 Anisotropic H elements Degenerate quadrilateral elements Anisotropic (degenerate) FEM: Application to problems with singularities Application to singularly perturbed problems Anisotropic finite element method behaves better in many practical problems! LSEC Shipeng M AO Anisotropic finite element methods with their applications 3/64
Numerical Simulation of the flow in the blood vessels (Sahni, Mueller [05,06]) Backgrounds Mathematical theory of anisotropic FEM Talk from Netwon’s interpolation polynomial From the viewpoint of Girault- Raviart interpolation From the viewpoint of orthogonal expansions 2 Anisotropic H elements Degenerate quadrilateral elements Application to problems with singularities Application to singularly perturbed problems Anisotropic FEM can save the computational cost significantly with the same accuracy as the isotropic FEM LSEC Shipeng M AO Anisotropic finite element methods with their applications 4/64
Anisotropic mesh for the inviscid flow around a supersonic jet (Bourgault et al [07]) Backgrounds Mathematical theory of anisotropic FEM Talk from Netwon’s interpolation polynomial From the viewpoint of Girault- Raviart interpolation From the viewpoint of orthogonal expansions 2 Anisotropic H elements Degenerate quadrilateral elements Application to problems with singularities Application to singularly perturbed problems LSEC Shipeng M AO Anisotropic finite element methods with their applications 5/64
Backgrounds Mathematical The application of anisotropic FEM theory of anisotropic FEM Talk from Netwon’s Many partial differential equations (PDEs) arising from science and engineering interpolation polynomial have a common feature that they have a small portion of the physical domain From the viewpoint of where small node separations are required to resolve large solution variations. Girault- Raviart interpolation From the Examples include problems having boundary layers, shock waves, ignition fronts, viewpoint of orthogonal and/or sharp interfaces in fluid dynamics, the combustion and heat transfer expansions 2 Anisotropic H theory, and groundwater hydrodynamics. elements Degenerate Numerical solution of these PDEs using a uniform mesh may be formidable when quadrilateral elements the systems involve more than two spatial dimensions since the number of mesh Application to nodes required can become very large. On the other hand, to improve efficiency problems with singularities and accuracy of numerical solution it is natural to put more mesh nodes in the Application to region of large solution variation than the rest of the physical domain. singularly perturbed problems The development of mathematical theory LSEC Shipeng M AO Anisotropic finite element methods with their applications 6/64
Backgrounds Mathematical theory of anisotropic FEM Talk from Netwon’s interpolation polynomial From the viewpoint of Mathematical theory of anisotropic FEM Girault- Raviart 2 interpolation Talk from Netwon’s interpolation polynomial From the viewpoint of orthogonal From the viewpoint of Girault- Raviart interpolation expansions 2 Anisotropic H From the viewpoint of orthogonal expansions elements Anisotropic H 2 elements Degenerate quadrilateral elements Application to problems with singularities Application to singularly perturbed problems LSEC Shipeng M AO Anisotropic finite element methods with their applications 7/64
Backgrounds Mathematical theory of anisotropic Some works FEM Talk from Netwon’s interpolation Mathematical studies of anisotropic meshes can be traced back to Synge [1957]. polynomial From the viewpoint of Error estimates of the continuous linear finite element with the maximal angle Girault- Raviart interpolation condition: Feng [1975], Babuska, Aziz [1976], Gregory[1975], Barnhill, Gregory From the viewpoint of [1976a,1976b], Jamet [1976], Oganesyan, Rukhovets[1979]. orthogonal expansions 2 Anisotropic H Until 90’s in the last century, especially in recent years, much attention is paid to elements Degenerate anisotropic FEM, Apel, Nicaise [92,99], Becker [95], Chen et al.[04], Duran [99], quadrilateral [ICM06] report...... elements Application to Most papers are on one special (or type of) finite element by studying its problems with singularities interpolation operator. Application to singularly We want to analyze it in a unified way through three new viewpoints perturbed problems LSEC Shipeng M AO Anisotropic finite element methods with their applications 8/64
Backgrounds Mathematical theory of anisotropic FEM Talk from Netwon’s interpolation polynomial From the viewpoint of Girault- Raviart interpolation From the Talk from Netwon’s interpolation polynomial viewpoint of orthogonal expansions 2 Anisotropic H elements Degenerate quadrilateral elements Application to problems with singularities Application to singularly perturbed problems LSEC Shipeng M AO Anisotropic finite element methods with their applications 9/64
Special property of the divided difference Backgrounds Lemma 1 Mathematical theory of anisotropic Let x 0 < x 1 < · · · < x m , be a uniform partition, d = x i + 1 − x i , 0 ≤ i ≤ m − 1 . Suppose FEM f ( x ) is sufficiently smooth, then Talk from Netwon’s interpolation Z x 1 Z t 1 + d Z t m − 1 + d polynomial 1 From the f ( m ) ( t m ) d t m . viewpoint of f [ x 0 , · · · , x m ] = d t 1 d t 2 · · · (1) Girault- Raviart m ! d m interpolation x 0 t 1 t m − 1 From the viewpoint of orthogonal expansions Remark Lemma 1 is similar to Hermite-Gennochi Theorem. 2 Anisotropic H elements Degenerate Theorem 2 quadrilateral elements Application to For all 0 ≤ l ≤ m , f [ x 0 , · · · , x m ] can be expressed by problems with singularities m − l Application to X singularly f [ x 0 , · · · , x m ] = c i f [ x i , · · · , x i + l ] , (2) perturbed problems i = 0 where c i ( 0 ≤ i ≤ m − l ) is only dependent to l and d. LSEC Shipeng M AO Anisotropic finite element methods with their applications 10/64
Rectangular elements of arbitrary order The interpolation polynomial I f ( x ) of f ( x ) satisfying I f ( x i ) = f ( x i ) , 0 ≤ i ≤ m , can Backgrounds be expressed in the following form Mathematical theory of anisotropic m i − 1 X Y FEM I f ( x ) = f [ x 0 , · · · , x i ] ( x − x j ) , (3) Talk from Netwon’s interpolation i = 0 j = 0 polynomial From the viewpoint of where p i ( x ) ( 0 ≤ i ≤ m ) ∈ P m and p i ( x j ) = δ ij , 0 ≤ i , j ≤ m . Girault- Raviart interpolation From the Denote the reference element ˆ K = [ 0 , 1 ] 2 , d = 1 / k , k is a positive integer, viewpoint of orthogonal expansions y ) ∈ C (ˆ x i = ˆ ˆ y i = id , i = 0 , · · · , k . Suppose ˆ u (ˆ x , ˆ K ) , then bi- k -interpolation 2 Anisotropic H elements polynomial ˆ u satisfying ˆ I ˆ u of ˆ I ˆ u (ˆ x i , ˆ y j ) = ˆ u (ˆ x i , ˆ y j )( 0 ≤ i , j ≤ k ) has the following Degenerate expression quadrilateral elements k k X X Application to ˆ I ˆ u (ˆ ˆ x i , ˆ y j ) ˆ p i (ˆ x ) ˆ p j (ˆ u = y ) , problems with singularities i = 0 j = 0 Application to singularly p i ( t ) ∈ P k (ˆ where ˆ K ) , ˆ p i (ˆ x l ) = ˆ p i (ˆ perturbed y l ) = δ il , 0 ≤ i , l ≤ k . Obviously problems ˆ I ˆ u = ˆ u , ∀ ˆ u ∈ Q k , (4) where Q k is the polynomial space of the degree ≤ k with respect to each variable. LSEC Shipeng M AO Anisotropic finite element methods with their applications 11/64
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