3.1.3  Discretization

The basics of Galerkin’s method consist of a clever discretization of the infinite linear space V  in the problem (3.5). The purpose of such a process is to create a controlled sub-space inside V  , which can be designed to approximate the solution of (3.6) with arbitrary precision.

Consider the finite sub-space Vh  of V  with dimension N. Consider also the functions ϕj=1...N  as an orthogonal basis of Vh  . Hence, every function v  in Vh  can be written as linear combination of the basis as in

    N
v = ∑  C ϕ ,v ∈ V  ⊂ V .
    j=1  j j      h
(3.9)

The discrete formulation of the variational problem (3.5) can be written as

                      ∫           ∫
Find u  ∈ V  such that    u′v′dx =   fvdx     ∀v ∈ V  .
      h    h              h                        h
(3.10)

Actually, the formulation (3.10) can be simplified. It can be proven that if uh  satisfies (3.10) only for the basis function of Vh  , it satisfies it for every element in Vh  . To show this, replace v  in (3.10) by the basis projection (3.9) as in

               ∫           ∫
                  u′hv′dx =   fvdx  ,

∫     N∑             ∫   ∑N
   u′h(   Cjϕj)′dx =   f    Cjϕjdx  ,
      j=1                j=1
  ∑N    ∫           ∑N    ∫
      Cj   u′ϕ′dx =    Cj    fϕjdx .                 (3.11)
  j=1       h j     j=1

On the assumption ∫ u ′ϕ′= ∫ fϕ
   h j       j  for j = 1..N, the equality (3.11) holds and u
 h  satisfies (3.10) for every element of Vh  . Consequently, the simplified version of (3.10) can be summarized by:

                     ∫           ∫
find u  ∈ V  such that    u′ϕ′dx =    fϕ dx,j = 1..N  .
     h    h              h j          j
(3.12)

The formulation (3.12) is well suitable for the construction of a numerical scheme, because it restricts the search for the solution uh  to the computation of a finite number of equations (N). Hence, the problem shifts to a more concrete perspective, and a method is required to compute uh  from those N equations. For that purpose, consider the expansion of u
 h  in the basis of V
 h  , in the same way as was done before for v
 h  (3.9). By substituting v
 h  appropriately in (3.10) the obtained function is

       ∫   ′ ′     ∫
          uhϕjdx =   f ϕjdx,j = 1..N  ,
∫                  ∫
   ∑N      ′ ′
  (    Ciϕi)ϕjdx =   f ϕjdx,j = 1..N  ,
   i=1
  ∑N    ∫  ′ ′     ∫
     Ci   ϕiϕjdx =   f ϕjdx,j = 1..N  .              (3.13)
  i=1

In fact, the substitution of uh  expansion in (3.12) leads to the transformation of the discrete problem in a linear system Ax = b  with N equations, where A, x and b are given by

    ⌊ ∫              ∫        ⌋       ⌊   ⌋      ⌊ ∫       ⌋
        ϕ1ϕ1dx  ⋅⋅⋅    ϕ1ϕN dx         C1            fϕ1dx
A = |⌈     ...      ...      ...    |⌉ , x = |⌈ ... |⌉ , b = |⌈   ...    |⌉ .
      ∫              ∫                             ∫
       ϕN ϕ1dx  ⋅⋅⋅    ϕNϕN dx         CN           f ϕNdx
(3.14)