8.5 The Galerkin Method
471
Equations (8.5.4) and (8.5.5) are both called the residual function. Obviously if
u n in Eqs. (8.5.4) and (8.5.5) is the analytic solution u, then the residuals R v and R s
should be equal to zero.
The basic idea of the weighted residual method is: Properly choosing two functions W vi and W si , they are called the weighted function in domain and boundary
weighted function (weighted function on boundary) respectively, also called the
weighted function, weight function or weighting function for short, such that the
products of the residuals R v and R s and their corresponding weighted functions are
equal to zero in a certain sense, namely putting the inner products of the residuals
and weighted functions satisfy the orthogonal conditions
(R v , W vi ) v =
V
R v W vi dV =
V
(T u n − f )W vi dV = 0
(8.5.6(a))
(R s , W si ) s =
S
R s W si dS =
S
(Bu n − g)W si dS = 0
(8.5.6(b))
where, (R v , W vi ) v and (R s , W si ) s are called the inner product in domain and
boundary inner product (inner product on boundary) respectively, they are called
the inner product for short.
If properly choosing the function of approximate solution u n , such that it satisfies
the boundary condition (8.5.2), then the inner product (8.5.6(a)) reduces to
(R v , W vi ) v =
V
R v W vi dV =
V
(T u n − f )W vi dV = 0
(8.5.7)
At this point, the inner product (8.5.7) is called the internal method of weighted
residual method.
If the function of approximate solution u n satisfies the operator Eq. (8.5.1), then
the inner product (8.5.6(b)) reduces to
(R s , W si ) s =
S
R s W si dS =
S
(Bu n − g)W si dS = 0
(8.5.8)
At this point, the inner product (8.5.8) is called the boundary method of weighted
residual method.
If the approximate function u n neither satisfy the operator Eq. (8.5.1), nor
satisfy the boundary condition (8.5.2), the inner product (8.5.6(a)) and inner product
(8.5.6(b)) must simultaneously be applied to eliminate the residual, then which are
called the mixed method of weighted residual method.
Substituting the approximate solution (8.5.3) into the inner product (8.5.6(a)),
and choosing n weighted functions W vi (i = 1, 2, . . . , n), unknown of the algebraic
equations with unknown quantities a i (i = 1, 2, . . . , n) can be established
471
Equations (8.5.4) and (8.5.5) are both called the residual function. Obviously if
u n in Eqs. (8.5.4) and (8.5.5) is the analytic solution u, then the residuals R v and R s
should be equal to zero.
The basic idea of the weighted residual method is: Properly choosing two functions W vi and W si , they are called the weighted function in domain and boundary
weighted function (weighted function on boundary) respectively, also called the
weighted function, weight function or weighting function for short, such that the
products of the residuals R v and R s and their corresponding weighted functions are
equal to zero in a certain sense, namely putting the inner products of the residuals
and weighted functions satisfy the orthogonal conditions
(R v , W vi ) v =
V
R v W vi dV =
V
(T u n − f )W vi dV = 0
(8.5.6(a))
(R s , W si ) s =
S
R s W si dS =
S
(Bu n − g)W si dS = 0
(8.5.6(b))
where, (R v , W vi ) v and (R s , W si ) s are called the inner product in domain and
boundary inner product (inner product on boundary) respectively, they are called
the inner product for short.
If properly choosing the function of approximate solution u n , such that it satisfies
the boundary condition (8.5.2), then the inner product (8.5.6(a)) reduces to
(R v , W vi ) v =
V
R v W vi dV =
V
(T u n − f )W vi dV = 0
(8.5.7)
At this point, the inner product (8.5.7) is called the internal method of weighted
residual method.
If the function of approximate solution u n satisfies the operator Eq. (8.5.1), then
the inner product (8.5.6(b)) reduces to
(R s , W si ) s =
S
R s W si dS =
S
(Bu n − g)W si dS = 0
(8.5.8)
At this point, the inner product (8.5.8) is called the boundary method of weighted
residual method.
If the approximate function u n neither satisfy the operator Eq. (8.5.1), nor
satisfy the boundary condition (8.5.2), the inner product (8.5.6(a)) and inner product
(8.5.6(b)) must simultaneously be applied to eliminate the residual, then which are
called the mixed method of weighted residual method.
Substituting the approximate solution (8.5.3) into the inner product (8.5.6(a)),
and choosing n weighted functions W vi (i = 1, 2, . . . , n), unknown of the algebraic
equations with unknown quantities a i (i = 1, 2, . . . , n) can be established