This brief write up attempts to explain the physical interpretation of Galerkin's weighted residual statement in the context of formulation of finite element equations.
As per the Galerkin's weighted residual statement the integration of the product of the error function in the finite element solution and a trial function is equated to zero.
'N' such equations are formed where 'N' equals the number of trial functions.
Now, taking the inner product of the two functions R and N equal to zero implies that these functions are orthogonal hence 'R' has no component along N
Since we form such equations for the whole set of trial functions, it implies that 'R' is orthogonal to all the trial functions.
And since the solution is comprised of trial functions, making the error orthogonal to all the trial functions implies that error is orthogonal to the solution space hence has no component along the solution space.
And this is what the Galerkin's weighted residual method does: makes the error vanish along the solution space!
As per the Galerkin's weighted residual statement the integration of the product of the error function in the finite element solution and a trial function is equated to zero.
'N' such equations are formed where 'N' equals the number of trial functions.
Now, taking the inner product of the two functions R and N equal to zero implies that these functions are orthogonal hence 'R' has no component along N
Since we form such equations for the whole set of trial functions, it implies that 'R' is orthogonal to all the trial functions.
And since the solution is comprised of trial functions, making the error orthogonal to all the trial functions implies that error is orthogonal to the solution space hence has no component along the solution space.
And this is what the Galerkin's weighted residual method does: makes the error vanish along the solution space!
