A Gradient Recovery Operator Based on an Oblique Projection

Current Research ProjectsFinite Element Method

Construction of the Gradient Recovery Operator

Bishnu P. Lamichhane, [email protected]

School of Mathematical and Physical Sciences, University of Newcastle, Australia

CARMA RetreatJuly 18, 2011

Bishnu P. Lamichhane, [email protected] A Gradient Recovery Operator Based on an Oblique Projection

Table of Contents

1 Current Research Projects

2 Finite Element Method

3 Construction of the Gradient Recovery OperatorOrthogonal ProjectionOblique Projection

Current Research Projects

My current research projects include

Solid Mechanics

A symmetric mixed finite element method for nearly incompressible elasticity based onbiorthogonal 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 DifferentialEquations, DOI: 10.1002/num.20683.

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 atthe ANU. A part of this project is finished in 2011 and is accepted to appear in ANZIAMJournal: electronic supplement.

My current research projects include

Solid Mechanics

A symmetric mixed finite element method for nearly incompressible elasticity based onbiorthogonal 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 DifferentialEquations, DOI: 10.1002/num.20683.

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 atthe ANU. A part of this project is finished in 2011 and is accepted to appear in ANZIAMJournal: electronic supplement.

Solid Mechanics

A finite element method based on simplices for the Hu-Washizu formulation in linearelasticity.In collaboration with Prof. B. Daya Reddy, University of Cape Town, South Africa. Closeto 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.

Solid Mechanics

Image Processing

Solid Mechanics

Image Processing

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 onNumerical Analysis, Vol 37 (2010), 166–172.

Biharmonic Problem ∆2u = 0

A mixed finite element method for the biharmonic problem using biorthogonal orquasi-biorthogonal systems.This project was finished in 2009 and is now published in Journal of Scientific Computing,Vol 46 (2011), 379–396.

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 onNumerical Analysis, Vol 37 (2010), 166–172.

A mixed finite element method for the biharmonic problem using biorthogonal orquasi-biorthogonal systems.This project was finished in 2009 and is now published in Journal of Scientific Computing,Vol 46 (2011), 379–396.

A Stabilized Mixed Finite Element Method for the Biharmonic Equation Based onBiorthogonal Systems.This project was finished in 2010 and is now published in Journal of Computational andApplied Mathematics, DOI:10.1016/j.cam. 2011.05.005. Future Goal: time-dependentproblems and improve the estimate.

Solid Mechanics

Two Finite Element Methods for Nearly Incompressible Linear Elasticity Using SimplicialMeshes, Submitted Book Chapter for Nova Science Publisher, New York.

A Stabilized Mixed Finite Element Method for the Biharmonic Equation Based onBiorthogonal Systems.This project was finished in 2010 and is now published in Journal of Computational andApplied Mathematics, DOI:10.1016/j.cam. 2011.05.005. Future Goal: time-dependentproblems and improve the estimate.

Solid Mechanics

Two Finite Element Methods for Nearly Incompressible Linear Elasticity Using SimplicialMeshes, Submitted Book Chapter for Nova Science Publisher, New York.

Finite Element Method

The main idea: For a problem posed in a continuous space W , we replace the continuousspace W (which is infinite dimensional) by a discrete one W h. For example, for Ω ⊂ Rd,d ∈ 1, 2, 3, the continuous space

H1(Ω) = u ∈ L2(Ω), ∇u ∈ [L2(Ω)]d

can be replaced by a discrete space

W h = spanφ1, φ2, · · · , φn.

A hanging node

Fiinite element basis functions in 1D

A finite element basis function in 2D

φi A hanging node

Fiinite element basis functions in 1D

A finite element basis function in 2D

Wh ⊂ H1(Ω) if there are no hanging nodes (right picture).

Piecewise Linear Space: A Finite Element Space

Let ∆ = a = x0 < x1 < · · · < xn = bbe a set of points in [a, b], and T =Iin−1

i=0 , where Ii = [xi, xi+1). Thepiecewise linear interpolant is obtained byjoining the set of data points

(x0, f(x0)), (x1, f(x1)), · · · , (xn, f(xn))

by a series of straight lines, as shown inthe adjacent figure.

Linear Space Sl(∆)

Let Sl(∆) be the space of all piecewise linear functions with respect to ∆. One canshow that Sl(∆) is a linear space of dimension n+ 1.

This is like a rational approximation of an irrational number like π.

Piecewise Linear Space: A Finite Element Space

Let ∆ = a = x0 < x1 < · · · < xn = bbe a set of points in [a, b], and T =Iin−1

i=0 , where Ii = [xi, xi+1). Thepiecewise linear interpolant is obtained byjoining the set of data points

(x0, f(x0)), (x1, f(x1)), · · · , (xn, f(xn))

by a series of straight lines, as shown inthe adjacent figure.

Linear Space Sl(∆)

Let Sl(∆) be the space of all piecewise linear functions with respect to ∆. One canshow that Sl(∆) is a linear space of dimension n+ 1.

This is like a rational approximation of an irrational number like π.

Piecewise Linear Space: Basis Functions

We can form a basis of Sh = Sl(∆). Define

φ0(x) =

x− x1

x0 − x1if x ∈ I0

0 otherwise, φn(x) =

x− xn−1

xn − xn−1if x ∈ In−1

0 otherwiseand

φi(x) =

x− xi−1

xi − xi−1if x ∈ Ii−1

x− xi+1

xi − xi+1if x ∈ Ii

0 otherwise

, for i = 1, . . . , n− 1,

Then φini=0 forms a basis for the space Sl(∆). A piecewise linear interpolant of acontinuous function u is written as Ihu ∈ Sl(∆) with

Ihu(x) =

n∑i=0

u(xi)φi(x).

φiφ0 φn

Orthogonal ProjectionOblique Projection

The Considered Problem

Weak Derivative

The function Ihu is continuous but not differentiable at the nodes x0, x1, · · · , xn.The weak derivative of Ihu is a piecewise constant function, but it is not continuous.

Generically, we can think of a finite element space Sh consisting of piecewisepolynomials for Ω ⊂ Rd:

Sh := vh ∈ C0(Ω) : vh|T ∈ Pp(T ), T ∈ Th, p ∈ N,

where T is a simplex, and Pp(T ) is the space of polynomials of total degree less thanor equal to p in T .

Orthogonal Projection of Weak Derivative

We now want to project the gradient of Ihu onto the space Sh. The projected gradient isthen continuous. But what do we gain? In fact, the projected gradient approximates theexact gradient ∇u better than the unprojected discrete gradient ∇Ihu.

The Considered Problem

Weak Derivative

The function Ihu is continuous but not differentiable at the nodes x0, x1, · · · , xn.The weak derivative of Ihu is a piecewise constant function, but it is not continuous.

Generically, we can think of a finite element space Sh consisting of piecewisepolynomials for Ω ⊂ Rd:

Sh := vh ∈ C0(Ω) : vh|T ∈ Pp(T ), T ∈ Th, p ∈ N,

where T is a simplex, and Pp(T ) is the space of polynomials of total degree less thanor equal to p in T .

Orthogonal Projection of Weak Derivative

We now want to project the gradient of Ihu onto the space Sh. The projected gradient isthen continuous. But what do we gain? In fact, the projected gradient approximates theexact gradient ∇u better than the unprojected discrete gradient ∇Ihu.

Project a Component of the Gradient Vector

Let ∇Ihu = 〈g1h, · · · , gdh〉 and consider a component of the gradient vector g1

h = ∂Ihu∂x1

Let p1h be its projection onto the space Sh, then p1

h ∈ Sh is defined as∫Ω

p1hφj dx =

g1hφj dx, φj ∈ Sh.

Since p1h ∈ Sh, we have p1

h =∑n

i=1 ciφi, for some constants c1, · · · , cn.

Linear System

Putting this expression back in the projection formula, we get Mc = r where M is amatrix with (i, j)th entry

φiφj dx,

c and r are vectors of length n with the jth component cj and∫

Ωg1hφj dx. Here M is a

mass matrix, which is sparse. We need to invert this matrix to computer c. In higherdimension, it is not easy to invert.

Project a Component of the Gradient Vector

Let ∇Ihu = 〈g1h, · · · , gdh〉 and consider a component of the gradient vector g1

h = ∂Ihu∂x1

Let p1h be its projection onto the space Sh, then p1

h ∈ Sh is defined as∫Ω

p1hφj dx =

g1hφj dx, φj ∈ Sh.

Since p1h ∈ Sh, we have p1

h =∑n

i=1 ciφi, for some constants c1, · · · , cn.

Linear System

Putting this expression back in the projection formula, we get Mc = r where M is amatrix with (i, j)th entry

φiφj dx,

c and r are vectors of length n with the jth component cj and∫

Ωg1hφj dx. Here M is a

mass matrix, which is sparse. We need to invert this matrix to computer c. In higherdimension, it is not easy to invert.

Super-Approximation Property

Let Rh be the component-wise projection operator onto Sh. For the projected gradientRh∇Ihu, we can show that (for good meshes)

‖∇u−Rh∇Ihu‖0,Ω ≤ Ch1+ 12 ‖u‖W3,∞(Ω),

whereas‖∇u−∇Ihu‖0,Ω ≤ Ch1+0‖u‖W3,∞(Ω).

Oblique Projection

Gradient projection gives a better estimate but expensive to compute. What happens ifwe replace the orthogonal projection Rh∇Ihu by an oblique projection Qh∇Iuu? Theoblique projection p1

h ∈ Sh of the first component of the gradient ∇Ihu is obtained by∫Ω

p1hµj dx =

g1hµj dx, µj ∈Mh.

This also gives a linear system Mc = r, but the (i, j)th component of M is defined as

φiµj dx.Bishnu P. Lamichhane, [email protected] A Gradient Recovery Operator Based on an Oblique Projection

Super-Approximation Property

Let Rh be the component-wise projection operator onto Sh. For the projected gradientRh∇Ihu, we can show that (for good meshes)

‖∇u−Rh∇Ihu‖0,Ω ≤ Ch1+ 12 ‖u‖W3,∞(Ω),

whereas‖∇u−∇Ihu‖0,Ω ≤ Ch1+0‖u‖W3,∞(Ω).

Oblique Projection

Gradient projection gives a better estimate but expensive to compute. What happens ifwe replace the orthogonal projection Rh∇Ihu by an oblique projection Qh∇Iuu? Theoblique projection p1

h ∈ Sh of the first component of the gradient ∇Ihu is obtained by∫Ω

p1hµj dx =

g1hµj dx, µj ∈Mh.

This also gives a linear system Mc = r, but the (i, j)th component of M is defined as

φiµj dx.Bishnu P. Lamichhane, [email protected] A Gradient Recovery Operator Based on an Oblique Projection

Diagonal Matrix M

Can we define the space Mh such that the matrix M is diagonal?Let the space of the standard finite element functions Sh be spanned by the basisφ1, . . . , φn. We construct the basis µ1, . . . , µn of the space Mh of test functions sothat the basis functions of Sh and Mh satisfy a condition of biorthogonality relation∫

µi φj dx = cjδij , cj 6= 0, 1 ≤ i, j ≤ n, (1)

where n := dimMh = dimSh, δij is the Kronecker symbol, and cj a scaling factor, andis always positive.

Oblique vs Orthogonal Projection

It is easy to verify that Qh is a projection onto the spaceSh. We note that Qh is not the orthogonal projectiononto Sh but an oblique projection onto it. Refer to[Gal03, Szy06] for more detail.

Orthogonal projection is unique, but the obliqueprojection depends on a direction.

Projection of OP onto OX

Y Orthogonal: OR

Oblique along PS: OS

Diagonal Matrix M

Can we define the space Mh such that the matrix M is diagonal?Let the space of the standard finite element functions Sh be spanned by the basisφ1, . . . , φn. We construct the basis µ1, . . . , µn of the space Mh of test functions sothat the basis functions of Sh and Mh satisfy a condition of biorthogonality relation∫

µi φj dx = cjδij , cj 6= 0, 1 ≤ i, j ≤ n, (1)

where n := dimMh = dimSh, δij is the Kronecker symbol, and cj a scaling factor, andis always positive.

Oblique vs Orthogonal Projection

It is easy to verify that Qh is a projection onto the spaceSh. We note that Qh is not the orthogonal projectiononto Sh but an oblique projection onto it. Refer to[Gal03, Szy06] for more detail.

Orthogonal projection is unique, but the obliqueprojection depends on a direction.

Projection of OP onto OX

Y Orthogonal: OR

Oblique along PS: OS

Properties of the Gradient Recovery Operator

The following lemma establishes the approximation property of operator Qh for a functionv ∈ H1+s(Ω).

For a function v ∈ Hs+1(Ω), s > 0, there exists a constant C independent of themesh-size h so that

‖v −Qhv‖L2(Ω) ≤ Ch1+r|v|Hr+1(Ω)

‖v −Qhv‖H1(Ω) ≤ Chr|v|Hr+1(Ω),(2)

where r := mins, p.

Properties of the Gradient Recovery Operator

We show that operator Qh has the same approximation property as the orthogonalprojection 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 errorestimation.

Theorem

We have

‖∇Ihu− Ph∇Ihu‖L2(Ω) ≤ ‖∇Ihu−Qh∇Ihu‖L2(Ω) ≤1

β‖∇Ihu− Ph∇Ihu‖L2(Ω). (3)

where β > 0 and Ph is the vector version of the L2-projection operator Ph.

The Gradient Recovery Operator: Applications

Since Sh and Mh form a biorthogonal system, we can write Qh as

n∑i=1

∫Ωµi v dx

ciφi (locality). (4)

If v has compact support, then Qhv has also compact support.

Since the error estimator based on an orthogonal projection is asymptotically exactfor mildly unstructured meshes, the error estimator based on this oblique projectionis 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 iseasy and cheap to compute.

The error estimator on element T is defined as

ηT = ‖Qh∇uh −∇uh‖L2(T ),

where uh is the finite element solution of some boundary value problem.

A. Galantai.Projectors and Projection Methods.Kluwer Academic Publishers, Dordrecht, 2003.

D.B. Szyld.The many proofs of an identity on the norm of oblique projections.Numerical Algorithms, 42:309–323, 2006.

A Gradient Recovery Operator Based on an Oblique Projection

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