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4.3 The Finite Element Method     continued...

The formulation of the finite element approximation starts with the Galerkin approximation, , where is our test function. We now use the finite element method to turn the continuous problems into a discrete formulation. First we discretize the solution domain, , and define a finite dimensional subspace, . One usually defines parameters of the function at node points, , i = 0, 1, ..., N. If we now define the basis functions, , as linear continuous piecewise functions that take the value 1 at node points and zero at other node points, then we can represent the function as:

such that each can be written in a unique way as a linear combination of the basis functions . Now the finite element approximation of the original boundary value problem can be stated as: