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4.1 The Galerkin Method - Optional     continued...

To express our problem in a Galerkin form, we begin by rewriting (1), as:

where A is the differential operator, . An equivalent statement of (15) is, find such that . Here, is an arbitrary test function, which can be thought of physically as a virtual potential field, and the notation, , denotes the inner product in , i.e. the space of square integrable functions. Applying Green's theorem, we can equivalently write,

where the notation denotes the inner product on the boundary S. When the Dirichlet, , and Neumann, , boundary conditions are specified on S, we obtain the weak form of (1):