Mathematical Physics - Volume II - Numerical Methods

Chapter 2. Finite element method

46

where ˜ u ( z ) is a smooth function of z . Finally, if time is limited by 0 ≤ t ≤ T , we can define the boundary problem as:

pizz k ( z )

∂ z

∂ u ( z , t )

∂ u ( z , t )

C ( z )

= f ( z , t ) 0 ≤ z ≤ l 0 < t ≤ T

∂ t −

(2.101)

u ( 0 , t ) = u ( l , t ) = 0 0 ≤ t ≤ T u ( z , 0 ) = ˜ u ( z ) 0 < z < l .

Of many ways in which variation formulation can be applied to problem (2.101), we will focus on one of the simpler ones. For this purpose, we will choose a favourable smooth function v = v ( z ) , independent from t , and we will multiply equation (2.99) with it. Then, after partial integration (see (2.25)), we obtain: Z l 0 C ( z ) ∂ u ( z , t ) ∂ t v ( z )+ k ( z ) ∂ U ( z , t ) ∂ z ∂ v ( z ) ∂ z d z − Z l 0 f ( z , t ) v ( z ) d z = 0 . (2.102) Equation (2.102) holds for ∀ v ( z ) ∈ H 1 0 ( 0 , l ) , and its solution should be found for u = u ( z , t ) ∈ H 1 0 ( 0 , l ) for 0 ≤ t ≤ T . H 1 0 ( 0 , l ) is the usual space of allowable functions with square-integral derivatives for 0 < z < l , whose values are 0 for z = 0 and z = l . The method for solving of equation (2.102) is analogous to the previously described one, with one crucial difference: nodal values of an approximate solution u h are unknown functions of time. Hence we can write: u h ( z , t ) = N ∑ j = 1 u j ( t ) φ j ( z ) , (2.103) where φ j ( z ) are the base functions defining space H h , u j ( t ) is the value of u h in node z j and at time t

N ∑ i = 1

u i ( t ) φ i ( z j ) = u j ( t ) .

u h ( z j , t ) =

(2.104)

When applied to equation (2.102), Galerkin method provides a system of N ordinary differential equations with a total of N unknown functios u j ( t ) :

d U ( t ) dt

C

+ KU ( t ) = f ( t ) 0 < t ≤ T U ( 0 ) = Uˆ ,

(2.105)

where

C i j = Z K i j = Z f i ( t ) = Z

l

C ( z ) φ i ( z ) φ j ( z ) d z

0

d φ j ( z ) dz

d φ i ( z ) dz

l

(2.106)

K ( z )

d z

0

l

f ( z , t ) φ i ( z ) d z .

0

Vector U , with an order of N × 1, made of nodal values N j ( t ) and U˜ are the interpolated values of ˜ U (i.e. the N × 1 vector of nodal values ˜ U ( z ) ). C is the heat capacity matrix, K is the typical conductivity matrix for a stationary problem and f is the time-independent load vector. Matrices C and K are sparse, with a narrow strip, symmetrical and invertible, and are obtained in the previously described manner. By applying finite element method, we obtained system of ordinary differential equations. Since u h was no discretized with respect tot time, equation (2.103) is only discretized in space. In order to