Global Projection Operator

We construct the Galerkin projection that minimizes the L₂ error for approximation of a function by the C⁰ basis. First we define the elemental Jacobi transform from the polynomial space to the physical space:

$$u = \sum^N \hat{u}_n \phi_n$$

where $\hat{u}_n$ is the coefficient of the n^th basis function $\phi_n$. From this we can obtain an approximation for the coefficients $\hat{u}_n$. If we take the inner product of both side with $\phi_m$ we obtain:

$$(\phi_m, u) = \sum^N (\phi_m, \phi_n) \hat{u}_n,$$

which we can solve for $\hat{u}_n$ since $(\phi_m, \phi_n)$ is positive definite [20]. Explicitly, we can write the projection coefficients as:

$$\hat{u}_m = B^{-1}_{mn} (\phi_n, u)$$

where B is the matrix with entries $B_{mn} = (\phi_m, \phi_n)$.This approximation minimizes the residual:

$$r = \Vert u - \sum^N \hat{u}_m \phi_m \Vert _{L_2}.$$