Example 2: Poisson's Equation
Our aim is to solve Poisson's equation
on a domain with on its boundary, and we want to specify the solution space later. We will use the -scalar product
to derive our weak formulation. Then, testing with differentiable functions, we get
We can make the left side of this equation more symmetric by integration by parts using Green's identity:
This is what is usually called the weak formulation of Poisson's equation; what's missing is the space, which is beyond the scope of this article. The space must allow us to write down this equation. Therefore, we should require that the derivatives of functions in this space are square integrable. Now, there is actually the Sobolev space of functions with weak derivatives in and with zero boundary conditions, which fulfills this purpose.
We obtain the generic form by assigning
and
Read more about this topic: Weak Formulation
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