A Gradient Recovery Operator Based on an Oblique Projection

Transcription

1 A Gradient Recovery Operator Based on an Bishnu P. Lamichhane, School of Mathematical and Physical Sciences, University of Newcastle, Australia CARMA Retreat July 18, 2011

2 Table of Contents 1 Current Research Projects 2 3 Orthogonal Projection

3 Current Research Projects My current research projects include Solid Mechanics A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. A joint project with Prof. Ernst Stephan, Leibniz University Hanover, Hanover, Germany. This project is just finished and published in Numerical Methods for Partial Differential Equations, DOI: /num Data Smoothing Approximation of thin plates spline using a mixed finite element method. In collaboration with Prof. Markus Hegland, Dr. Stephen Roberts and Dr. Linda Stals at the ANU. A part of this project is finished in 2011 and is accepted to appear in ANZIAM Journal: electronic supplement.

4 Current Research Projects My current research projects include Solid Mechanics A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. A joint project with Prof. Ernst Stephan, Leibniz University Hanover, Hanover, Germany. This project is just finished and published in Numerical Methods for Partial Differential Equations, DOI: /num Data Smoothing Approximation of thin plates spline using a mixed finite element method. In collaboration with Prof. Markus Hegland, Dr. Stephen Roberts and Dr. Linda Stals at the ANU. A part of this project is finished in 2011 and is accepted to appear in ANZIAM Journal: electronic supplement.

5 Current Research Projects Solid Mechanics A finite element method based on simplices for the Hu-Washizu formulation in linear elasticity. In collaboration with Prof. B. Daya Reddy, University of Cape Town, South Africa. Close to be finished. Parameter Estimation of PDEs Parameter estimation in elliptic partial differential equations. This project is very close to be finished and is done with Dr. Robert S (Bob) Anderssen, CSIRO, ANU. Trying to submit for MODSIM proceeding. Image Processing Total variation minimisation in removing the mixture of Gaussian and impulsive noise. I am collaborating with Dr. David Allingham, CARMA, to complete this project.

6 Current Research Projects Solid Mechanics A finite element method based on simplices for the Hu-Washizu formulation in linear elasticity. In collaboration with Prof. B. Daya Reddy, University of Cape Town, South Africa. Close to be finished. Parameter Estimation of PDEs Parameter estimation in elliptic partial differential equations. This project is very close to be finished and is done with Dr. Robert S (Bob) Anderssen, CSIRO, ANU. Trying to submit for MODSIM proceeding. Image Processing Total variation minimisation in removing the mixture of Gaussian and impulsive noise. I am collaborating with Dr. David Allingham, CARMA, to complete this project.

7 Current Research Projects Solid Mechanics A finite element method based on simplices for the Hu-Washizu formulation in linear elasticity. In collaboration with Prof. B. Daya Reddy, University of Cape Town, South Africa. Close to be finished. Parameter Estimation of PDEs Parameter estimation in elliptic partial differential equations. This project is very close to be finished and is done with Dr. Robert S (Bob) Anderssen, CSIRO, ANU. Trying to submit for MODSIM proceeding. Image Processing Total variation minimisation in removing the mixture of Gaussian and impulsive noise. I am collaborating with Dr. David Allingham, CARMA, to complete this project.

8 Current Research Projects Finite Element Theory A gradient recovery operator based on an oblique projection. This project was finished in 2010 and is now published in Electronic Transactions on Numerical Analysis, Vol 37 (2010), Biharmonic Problem 2 u = 0 A mixed finite element method for the biharmonic problem using biorthogonal or quasi-biorthogonal systems. This project was finished in 2009 and is now published in Journal of Scientific Computing, Vol 46 (2011),

9 Current Research Projects Finite Element Theory A gradient recovery operator based on an oblique projection. This project was finished in 2010 and is now published in Electronic Transactions on Numerical Analysis, Vol 37 (2010), Biharmonic Problem 2 u = 0 A mixed finite element method for the biharmonic problem using biorthogonal or quasi-biorthogonal systems. This project was finished in 2009 and is now published in Journal of Scientific Computing, Vol 46 (2011),

10 Current Research Projects Biharmonic Problem 2 u = 0 A Stabilized Mixed for the Biharmonic Equation Based on Biorthogonal Systems. This project was finished in 2010 and is now published in Journal of Computational and Applied Mathematics, DOI: /j.cam Future Goal: time-dependent problems and improve the estimate. Solid Mechanics Two s for Nearly Incompressible Linear Elasticity Using Simplicial Meshes, Submitted Book Chapter for Nova Science Publisher, New York.

11 Current Research Projects Biharmonic Problem 2 u = 0 A Stabilized Mixed for the Biharmonic Equation Based on Biorthogonal Systems. This project was finished in 2010 and is now published in Journal of Computational and Applied Mathematics, DOI: /j.cam Future Goal: time-dependent problems and improve the estimate. Solid Mechanics Two s for Nearly Incompressible Linear Elasticity Using Simplicial Meshes, Submitted Book Chapter for Nova Science Publisher, New York.

12 The main idea: For a problem posed in a continuous space W, we replace the continuous space W (which is infinite dimensional) by a discrete one W h. For example, for Ω R d, d {1, 2, 3}, the continuous space can be replaced by a discrete space H 1 (Ω) = {u L 2 (Ω), u [L 2 (Ω)] d } W h = span{φ 1, φ 2,, φ n}. Fiinite element basis functions in 1D φ i T A finite element basis function in 2D φ i A hanging node T W h H 1 (Ω) if there are no hanging nodes (right picture).

13 Piecewise Linear Space: A Finite Element Space Let = {a = x 0 < x 1 < < x n = b} be a set of points in [a, b], and T = {I i} n 1 i=0, where Ii = [xi, xi+1). The piecewise linear interpolant is obtained by joining the set of data points {(x 0, f(x 0)), (x 1, f(x 1)),, (x n, f(x n))} by a series of straight lines, as shown in the adjacent figure. Linear Space S l ( ) Let S l ( ) be the space of all piecewise linear functions with respect to. One can show that S l ( ) is a linear space of dimension n + 1. This is like a rational approximation of an irrational number like π.

14 Piecewise Linear Space: A Finite Element Space Let = {a = x 0 < x 1 < < x n = b} be a set of points in [a, b], and T = {I i} n 1 i=0, where Ii = [xi, xi+1). The piecewise linear interpolant is obtained by joining the set of data points {(x 0, f(x 0)), (x 1, f(x 1)),, (x n, f(x n))} by a series of straight lines, as shown in the adjacent figure. Linear Space S l ( ) Let S l ( ) be the space of all piecewise linear functions with respect to. One can show that S l ( ) is a linear space of dimension n + 1. This is like a rational approximation of an irrational number like π.

15 Piecewise Linear Space: Basis Functions We can form a basis of S h = S l ( ). Define x x 1 if x I x x n 1 0 if x I n 1 φ 0(x) = x 0 x 1, φ n(x) = x n x n 1 0 otherwise 0 otherwise x x i 1 if x I i 1 x i x i 1 φ i(x) = x x i+1 if x I i, for i = 1,..., n 1, x i x i+1 0 otherwise and Then {φ i} n i=0 forms a basis for the space S l ( ). A piecewise linear interpolant of a continuous function u is written as I h u S l ( ) with n I h u(x) = u(x i)φ i(x). i=0 φ 0 φ i φn T

16 The Considered Problem Orthogonal Projection Weak Derivative The function I h u is continuous but not differentiable at the nodes {x 0, x 1,, x n}. The weak derivative of I h u is a piecewise constant function, but it is not continuous. Generically, we can think of a finite element space S h consisting of piecewise polynomials for Ω R d : S h := {v h C 0 (Ω) : v h T P p(t ), T T h }, p N, where T is a simplex, and P p(t ) is the space of polynomials of total degree less than or equal to p in T. Orthogonal Projection of Weak Derivative We now want to project the gradient of I h u onto the space S h. The projected gradient is then continuous. But what do we gain? In fact, the projected gradient approximates the exact gradient u better than the unprojected discrete gradient I h u.

17 The Considered Problem Orthogonal Projection Weak Derivative The function I h u is continuous but not differentiable at the nodes {x 0, x 1,, x n}. The weak derivative of I h u is a piecewise constant function, but it is not continuous. Generically, we can think of a finite element space S h consisting of piecewise polynomials for Ω R d : S h := {v h C 0 (Ω) : v h T P p(t ), T T h }, p N, where T is a simplex, and P p(t ) is the space of polynomials of total degree less than or equal to p in T. Orthogonal Projection of Weak Derivative We now want to project the gradient of I h u onto the space S h. The projected gradient is then continuous. But what do we gain? In fact, the projected gradient approximates the exact gradient u better than the unprojected discrete gradient I h u.

18 Orthogonal Projection Project a Component of the Gradient Vector Let I h u = gh, 1, gh d and consider a component of the gradient vector gh 1 = I hu x 1. Let p 1 h be its projection onto the space S h, then p 1 h S h is defined as p 1 hφ j dx = ghφ 1 j dx, φ j S h. Ω Since p 1 h S h, we have p 1 h = n i=1 ciφi, for some constants c1,, cn. Linear System Ω Putting this expression back in the projection formula, we get Mc = r where M is a matrix with (i, j)th entry M ij = φ iφ j dx, Ω c and r are vectors of length n with the jth component c j and Ω g1 hφ j dx. Here M is a mass matrix, which is sparse. We need to invert this matrix to computer c. In higher dimension, it is not easy to invert.

19 Orthogonal Projection Project a Component of the Gradient Vector Let I h u = gh, 1, gh d and consider a component of the gradient vector gh 1 = I hu x 1. Let p 1 h be its projection onto the space S h, then p 1 h S h is defined as p 1 hφ j dx = ghφ 1 j dx, φ j S h. Ω Since p 1 h S h, we have p 1 h = n i=1 ciφi, for some constants c1,, cn. Linear System Ω Putting this expression back in the projection formula, we get Mc = r where M is a matrix with (i, j)th entry M ij = φ iφ j dx, Ω c and r are vectors of length n with the jth component c j and Ω g1 hφ j dx. Here M is a mass matrix, which is sparse. We need to invert this matrix to computer c. In higher dimension, it is not easy to invert.

20 Orthogonal Projection Super-Approximation Property Let R h be the component-wise projection operator onto S h. For the projected gradient R h I h u, we can show that (for good meshes) u R h I h u 0,Ω Ch u W 3, (Ω), whereas u I h u 0,Ω Ch 1+0 u W 3, (Ω). Gradient projection gives a better estimate but expensive to compute. What happens if we replace the orthogonal projection R h I h u by an oblique projection Q h I uu? The oblique projection p 1 h S h of the first component of the gradient I h u is obtained by p 1 hµ j dx = ghµ 1 j dx, µ j M h. Ω Ω This also gives a linear system Mc = r, but the (i, j)th component of M is defined as M ij = φ iµ j dx. Bishnu P. Lamichhane, bishnu.lamichhane@newcastle.edu.au Ω A Gradient Recovery Operator Based on an

21 Orthogonal Projection Super-Approximation Property Let R h be the component-wise projection operator onto S h. For the projected gradient R h I h u, we can show that (for good meshes) u R h I h u 0,Ω Ch u W 3, (Ω), whereas u I h u 0,Ω Ch 1+0 u W 3, (Ω). Gradient projection gives a better estimate but expensive to compute. What happens if we replace the orthogonal projection R h I h u by an oblique projection Q h I uu? The oblique projection p 1 h S h of the first component of the gradient I h u is obtained by p 1 hµ j dx = ghµ 1 j dx, µ j M h. Ω Ω This also gives a linear system Mc = r, but the (i, j)th component of M is defined as M ij = φ iµ j dx. Bishnu P. Lamichhane, bishnu.lamichhane@newcastle.edu.au Ω A Gradient Recovery Operator Based on an

22 Orthogonal Projection Diagonal Matrix M Can we define the space M h such that the matrix M is diagonal? Let the space of the standard finite element functions S h be spanned by the basis {φ 1,..., φ n}. We construct the basis {µ 1,..., µ n} of the space M h of test functions so that the basis functions of S h and M h satisfy a condition of biorthogonality relation µ i φ j dx = c jδ ij, c j 0, 1 i, j n, (1) Ω where n := dim M h = dim S h, δ ij is the Kronecker symbol, and c j a scaling factor, and is always positive. Oblique vs Orthogonal Projection It is easy to verify that Q h is a projection onto the space S h. We note that Q h is not the orthogonal projection onto S h but an oblique projection onto it. Refer to [Gal03, Szy06] for more detail. Orthogonal projection is unique, but the oblique projection depends on a direction. Projection of OP onto OX Orthogonal: OR Oblique along PS: OS P O R S Y X

23 Orthogonal Projection Diagonal Matrix M Can we define the space M h such that the matrix M is diagonal? Let the space of the standard finite element functions S h be spanned by the basis {φ 1,..., φ n}. We construct the basis {µ 1,..., µ n} of the space M h of test functions so that the basis functions of S h and M h satisfy a condition of biorthogonality relation µ i φ j dx = c jδ ij, c j 0, 1 i, j n, (1) Ω where n := dim M h = dim S h, δ ij is the Kronecker symbol, and c j a scaling factor, and is always positive. Oblique vs Orthogonal Projection It is easy to verify that Q h is a projection onto the space S h. We note that Q h is not the orthogonal projection onto S h but an oblique projection onto it. Refer to [Gal03, Szy06] for more detail. Orthogonal projection is unique, but the oblique projection depends on a direction. Projection of OP onto OX Orthogonal: OR Oblique along PS: OS P O R S Y X

24 Orthogonal Projection Properties of the Gradient Recovery Operator The following lemma establishes the approximation property of operator Q h for a function v H 1+s (Ω). Lemma For a function v H s+1 (Ω), s > 0, there exists a constant C independent of the mesh-size h so that v Q h v L 2 (Ω) Ch 1+r v H r+1 (Ω) v Q h v H 1 (Ω) Ch r v H r+1 (Ω), (2) where r := min{s, p}.

25 Orthogonal Projection Properties of the Gradient Recovery Operator We show that operator Q h has the same approximation property as the orthogonal projection operator. However, the new operator is more efficient than the old one. Hence, it is ideal to use this operator as a gradient recovery operator in the a posteriori error estimation. Theorem We have I h u P h I h u L 2 (Ω) I h u Q h I h u L 2 (Ω) 1 β I hu P h I h u L 2 (Ω). (3) where β > 0 and P h is the vector version of the L 2 -projection operator P h.

26 Orthogonal Projection The Gradient Recovery Operator: Applications Since S h and M h form a biorthogonal system, we can write Q h as n µi v dx Ω Q h v = φ i (locality). (4) c i If v has compact support, then Q h v has also compact support. i=1 Since the error estimator based on an orthogonal projection is asymptotically exact for mildly unstructured meshes, the error estimator based on this oblique projection is also asymptotically exact for such meshes. However, the orthogonal projection is not local and hence also expensive to compute. Our new oblique projection thus gives a local gradient recovery operator, which is easy and cheap to compute. The error estimator on element T is defined as η T = Q h u h u h L 2 (T ), where u h is the finite element solution of some boundary value problem.

27 Orthogonal Projection A. Galántai. Projectors and Projection Methods. Kluwer Academic Publishers, Dordrecht, D.B. Szyld. The many proofs of an identity on the norm of oblique projections. Numerical Algorithms, 42: , 2006.