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I have an embarrassingly basic question that I am, well, embarrassed to be asking.

Let's say I want to solve Poisson's equation with Dirichlet boundary conditions on the domain $\Omega = [-1, 1]\times[-1, 1]$. I want to do so for a sinusoidal source,

$\nabla^2 u = \sin(x)\sin(y)$ where $u = u(x, y)$.

Is there a way to do this without using Fourier series? I must admit, I haven't been solving many PDE's analytically recently, and have been a bit spoiled by ODE's where I can simply guess the solution. I do feel, however, that this is a simple question with a simple answer that I should see right away.

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    Well, $u(x,y) = -\sin(x)\sin(y)/2$ is a solution by inspection, but what are your boundary conditions? – 2017-01-12
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    If you mean homogeneous Dirichlet boundary conditions ($u=0$) on the boundary, then the solution is a complicated Fourier series. If the source term was $\sin(\pi x)\sin(\pi y)$ instead, then the answer is much simpler. – 2017-01-12

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