functions {Nj} on the fine grid QJ,

Hence, if the solution on grid J - 1 is uJ-1 = J2j UJ-1Nj, then the interpolation operator Ij-1, mapping uJ-1 to uJ, is defined by:

Next, we must define the linear systems on the different grids. In the previous Chapter 2 we defined the weak formulation of the Poisson problem,

The weak formulation, introduced in Chapter 2 of (4.97)-(4.98) is: Find u e V such that

(Au, v) = (Vu, Vv), Vu, v e V, where V was some Hilbert space.

We are now able to define the linear systems to be solved approximately on the different grids by using the weak formulation on the different finite element spaces VJ. The linear systems are

The explicit expressions for the matrices and right-hand sides are as follows. Let {Nj } be the finite element basis functions that span VJ. Then

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