PRINCIPLES OF MIMETIC DISCRETIZATIONS OF DIFFERENTIAL

OPERATORSA Research Paper By:

Bochev, P.B. and Hyman, J.M.

Presented By:Md. Hasan Ansari(071/MSCS/655)

IntroductionPartial differential equations (PDEs) are ubiquitous in science

and engineering.Key step in its Numerical solution is discretization.

Replace PDEs by the system of Algebraic EquationsDiscretization must be compatible or mimetic

The fundamental properties of continuum equations must be retained or preserved

Eg. Divergence of a Curl must equal zero. ( ∇ · (∇ × u) = 0 ) Curl of the gradient = 0 ( ∇ × ∇f = 0 )

This paper focus on obtaining such mimetic discrete versions of the differential operators like Gradient, Divergence, Curl and Laplacian

IntroductionMethods Popularly used for compatible discretization:

Finite Difference Method (FDM)Finite Element Method (FEM) Finite Volume Method (FVM)

The paper provides a common framework for mimetic discretization using algebraic topology.

Main idea of the paper is the use of two basic mappings between differential forms and cochains:A Reduction Map R and A Reconstruction Map I

The choice of I results the methods FDM, FEM or FVM.

Differential formsApproach to multivariable calculus that is independent of

coordinates.The vector space for algebraic k-form denoted by: Lk 0-forms are scalar functions of the form f(x,y,z) Point

(Scalar)1-forms are expressions of the form

f(x,y,z)dx+g(x,y,z)dy+h(x,y,z)dz f(x,y,z)i+g(x,y,z)j+h(x,y,z)k Path (vector)

2-forms are expressions of the form f(x,y,z)dy∧dz+g(x,y,z)dz∧dx+h(x,y,z)dx∧dy Surface (Vector)

3-forms are expressions of the form f(x,y,z)dx ∧ dy ∧ dz Volume (Scalar)

Wedge Product ∧ : Lk x Ll Lk+l for k+l <= nSimilar to cross product BUT it results bivector

Differential formsThe Hodge star operator relates zero-forms with three-

forms and one-forms with two-forms according to the relationships:⋆1 = dx ∧ dy ∧ dz and dx = dy⋆ ∧dz, dy = dz⋆ ∧dx, dz = dx⋆ ∧dyAlso, = 1, so that dy⋆⋆ ⋆ ∧dz = dx To generalize : ⋆ Lk Ln-k

Exterior derivative ‘d’ : exterior derivative of a differential form of degree k is

a differential form of degree k + 1. ie. d : Lk Lk+1 such thatd(wk ∧ wl) = (dwk) ∧ wl + (−1)kwk ∧ (dwl) ; k + l < n

Also, exterior derivative satisfies - dd = 0 and give rise to exact sequence:

Called the De Rham Complex.

grad curl div

CochainsA k-Chain: formal linear combination of k-simplexesA k-simplex sk is an ordered collection [p0, . . . , pk] of (k

+ 1), k <= n distinct points in Rn

A set of k-chains is denoted by Ck.The boundary ∂ of a k-simplex is (k − 1)-chainThe collection {C0,C1,C2,C3} is called chain complex if

for any c∈Ck, ∂c∈Ck−1. This gives rise to an exact sequence

The dual Ck is the collection of all linear functionals on Ck. The elements of Ck are called k-cochains. < . , . > denote duality pairing such that

<a,∂c> = <da,c> , Where d : CkCk+1 is coboundary satisfying dd = 0 and forms

Where, ∂k : Ck+1Ck is boundary operatoron k-chains that follows ∂∂ = 0

Maps differential k-forms to k-cochains by using De Rham Map so our R is De Rham Map for reduction operation The map is defined by

Where, c∈Ck is k-cochain and w∈Lk(W) is a k-form

The mapping w Rw establishes discrete representation of k-forms

Reduction Map ‘R’

Reconstruction Map ‘I’operation I that serves as an approximate inverse

to R and translates the global information stored in Ck back to local representations.

I is flexible because of the many possible ways in which global data from Ck can be combined in a local field representation.

Two basic conditions:RI = id [I must be right inverse of R] (consistency

property) IR = id + O(hs) [I must be approximate left inverse of R]

(approximation Property)where s and h are positive real numbers that give the approximation order and the partition size in K, respectively.

Thank You