Lectures IIT Kanpur, India Lecture 1: @=@x and dx › math › workshop-and-conference › WMSEM ›...

Lectures IIT Kanpur, IndiaLecture 1: ∂/∂x and dx

A new meaning for familiar symbols

Marc [email protected]

Department of Aerospace EngineeringDelft University of Technology

23 March 2015

Marc Gerritsma (TUD) Mimetic discretizations Lecture 1 1 / 27

Overview Introduction

Outline of this lecture

This short course on mimetic spectral elements consists of 6 lectures:

Lecture 1: In this lecture we will review some basic concepts from differential geometry

Lecture 2: Generalized Stokes Theorem and geometric integration

Lecture 3: Connection between continuous and discrete quantities. The Reductionoperator and the reconstruction operator.

Lecture 4: The Hodge-? operator. Finite volume, finite element methods andleast-squares methods.

Lecture 5: Application of mimetic schemes to elliptic equations. Poisson and Stokesproblem

Lecture 6: Open research questions. Collaboration.

The Poisson problemMy favorite appetizer

Consider the Poisson equation in R3 given by

−∆ϕ = f .

with appropriate boundary conditions. We can write this as a first ordersystem

u = ∇ϕ exact!u = q approximation

−∇ · q = f exact!

These lectures will be mainly concerned why and how we candiscretize the ∇ and ∇· exactly.

−∆ϕ = f .

On the Mathematical Classification of PhysicalQuantitiesJ.C. Maxwell, 1871.

How can it be, that completely different physical models are described by the same mathematicalequation?

This is the question that James Clerk Maxwell raised in 1871 in his paper "On the MathematicalClassification of Physical Quantities"

Now we know that similarities in mathematical descriptions arise from the fact that all physicstakes place in the same physical space.

We will decompose ‘space’ into a topological part and a metric part

Space, functions & vectors

Description of spaceManifolds

Functions and vectors I

With the charts we can identifya point P ∈M with a k -tuple(x1, . . . , xk )

We can assign a function valuef (P) at the point P using thecoordinate charts

f (P) = f (ϕ−1(x1, . . . , xk )) .

A curve through the point P canparameterized by s ∈ R:

γ(s) = (γ1(s), . . . , γk (s)) ,

withγ(0) = P .

With the parametrization

γ(s) = (γ1(s), . . . , γk (s)) ,

we can find the tangent vectorv along γ at P

v|P = γ(0) = (γ1(s), . . . , γk (s)) .

Any smooth curve passingthrough the point P generates atangent vector at P

Every curve through P generates a tangent vector at P. Conversely,any tangent vector at the point P can be associated to a class ofcurves through the point P. The space of tangent vectors at the point Pis called the Tangent space at P, denoted by TPM

Functions and vectors II

If we have a k -dimensional manifold then the dimension of TPM isequal to k . A special basis for this vector space at P is obtained bytaking the coordinate curves

γ(s) = (x1, . . . , x i + s, . . . , xk ) ,

where (x1, . . . , xk ) are the local coordinates for the point P. Theassociated tangent vector is then given by

γ(s) = (0, . . . , 1︸︷︷︸i

, . . . ,0) .

This is special basis is called a coordinate basis and usually denotedby

∂x i

∣∣∣∣P

:= (0, . . . , 1︸︷︷︸i

, . . . ,0)

Functions and vectors II

If we have a k -dimensional manifold then the dimension of TPM isequal to k . A special basis for this vector space at P is obtained bytaking the coordinate curves

γ(s) = (x1, . . . , x i + s, . . . , xk ) ,

where (x1, . . . , xk ) are the local coordinates for the point P. Theassociated tangent vector is then given by

γ(s) = (0, . . . , 1︸︷︷︸i

, . . . ,0) .

This is special basis is called a coordinate basis and usually denotedby

∂x i

∣∣∣∣P

:= (0, . . . , 1︸︷︷︸i

, . . . ,0)

Functions and vectors III

So any tangent vector can be written as a linear combination of thecoordinate basis vectors

v = v1 ∂

∣∣∣∣P

+ . . .+ vk ∂

∣∣∣∣P

= v i ∂

∂x i

∣∣∣∣P

It might seem strange these basis functions, but partial derivativesbehave in the same way under transformations as basis functions.

Functions and vectors III

So any tangent vector can be written as a linear combination of thecoordinate basis vectors

v = v1 ∂

∣∣∣∣P

+ . . .+ vk ∂

∣∣∣∣P

= v i ∂

∂x i

∣∣∣∣P

It might seem strange these basis functions, but partial derivativesbehave in the same way under transformations as basis functions.

Functions and vectorsIntermezzo

The definition of vectors as tangent vectors does not work well in thediscrete setting, because a curve is a continuous set of points,whereas in the discrete setting we only have a finite discrete set ofpoints.An alternative is to consider vectors at a point as derivations:v is a linear map from continuous functions at a point to continuousfunctions at a point which satisfies

v(fg) := v(f ) · g + f · v(g)

see L. Tu, An Introduction to ManifoldsHere v(f ) is defined as

v(f ) := v1 ∂f∂x1

∣∣∣∣P

+ . . .+ vk ∂f∂xk

∣∣∣∣P

Functions and vectorsIntermezzo

The definition of vectors as tangent vectors does not work well in thediscrete setting, because a curve is a continuous set of points,whereas in the discrete setting we only have a finite discrete set ofpoints.An alternative is to consider vectors at a point as derivations:v is a linear map from continuous functions at a point to continuousfunctions at a point which satisfies

v(fg) := v(f ) · g + f · v(g)

see L. Tu, An Introduction to ManifoldsHere v(f ) is defined as

v(f ) := v1 ∂f∂x1

∣∣∣∣P

+ . . .+ vk ∂f∂xk

∣∣∣∣P

Covectors

The covector I

Since TPM is a linear vector space we have also the dual vectorspace consisting of all linear functionals acting on TPM. This dualspace is denoted by T ∗PM.

So for any α ∈ T ∗PM, we have

α : TPM→ R

such that for v,w ∈ TpM and λ, µ ∈ R, we have

α(λv + µw) = λα(v) + µα(w)

Covectors

The covector I

Since TPM is a linear vector space we have also the dual vectorspace consisting of all linear functionals acting on TPM. This dualspace is denoted by T ∗PM.

So for any α ∈ T ∗PM, we have

α : TPM→ R

such that for v,w ∈ TpM and λ, µ ∈ R, we have

α(λv + µw) = λα(v) + µα(w)

Covectors

The covector II

If we use the expansion of a vector v at a point P in terms of thecoordinate basis functions ∂/∂x i and apply the covector α we have

α(v) = α

(v1 ∂

∣∣∣∣P

+ . . .+ vk ∂

∣∣∣∣P

)= v1α

∣∣∣∣P

)+ · · ·+ vkα

∣∣∣∣P

)So it is sufficient to know how α acts on the basis vectors to

determine the action of α.

Covectors

The covector II

α(v) = α

(v1 ∂

∣∣∣∣P

+ . . .+ vk ∂

∣∣∣∣P

)= v1α

∣∣∣∣P

)+ · · ·+ vkα

∣∣∣∣P

Covectors

The covector II

α(v) = α

(v1 ∂

∣∣∣∣P

+ . . .+ vk ∂

∣∣∣∣P

)= v1α

∣∣∣∣P

)+ · · ·+ vkα

∣∣∣∣P

Covectors

The covector III

Consider the covector αi with the property

αi(∂

∂x j

1 if i = j

0 otherwise

For a coordinate basis, this covector αi is generally denoted by dx i∣∣P

Any covector αi can written as a (unique) linear combination of thesebasis covectors

α = α1dx1 + . . . αk dxk .

Covectors

The covector III

αi(∂

∂x j

1 if i = j

0 otherwise

α = α1dx1 + . . . αk dxk .

Covectors

The covector III

αi(∂

∂x j

1 if i = j

0 otherwise

α = α1dx1 + . . . αk dxk .

Covectors

The covector IV

With these definitions we deduce that

α(v) = 〈α,v〉 =k∑

αiv i .

The operation 〈α,v〉 is called duality pairing and should not beconfused with an inner-product.

Sofar we have not introduced the concept of metric, so notions oflength of a vector or the angle between two vectors does not exist(yet).

Covectors

The covector IV

α(v) = 〈α,v〉 =k∑

αiv i .

Covectors

The covector IV

α(v) = 〈α,v〉 =k∑

αiv i .

Covectors

Vector and covector fields

Sofar, everything was point-wise defined. We can extend this to theentire manifold by assuming that the expansion coefficients aresmooth functions of (x1, . . . , xk ).

v(x1, . . . , xk ) = v1(x1, . . . , xk )∂

∣∣∣∣P

+ . . .+ vk (x1, . . . , xk )∂

∣∣∣∣P

= v i(x1, . . . , xk )∂

∂x i

∣∣∣∣P

α(x1, . . . , xk ) = α1(x1, . . . , xk )dx1 + . . . αk (x1, . . . , xk )dxk .

Covectors

v(x1, . . . , xk ) = v1(x1, . . . , xk )∂

∣∣∣∣P

+ . . .+ vk (x1, . . . , xk )∂

∣∣∣∣P

= v i(x1, . . . , xk )∂

∂x i

∣∣∣∣P

Covectors

v(x1, . . . , xk ) = v1(x1, . . . , xk )∂

∣∣∣∣P

+ . . .+ vk (x1, . . . , xk )∂

∣∣∣∣P

= v i(x1, . . . , xk )∂

∂x i

∣∣∣∣P

Wedge product

The wedge product

A covector field is also called a differentiable 1-form. The wedgeproduct allows us to multiply to 1-forms to get a 2-form. Let α and β be2 1-forms, then α ∧ β is a 2-form.