Depends on which PDEs you want to solve and how you want to approach them.
Every approach that I know of will require linear algebra. You'll want to find a good matrix package for .NET, the best you can find, one that can handle sparse matricies efficiently.
Linear elliptic (steady state diffusion), parabolic (transient diffusion), and hyperbolic (F= MA dynamic) PDEs require slightly different approaches.
You can use classical finite difference, finite element (weighted residual), or boundary element (Green's functions) to create the system matrix you'd like to solve. General non-linear PDEs are probably best attacked using a finite element/weighted residual technique.
All these are big topics unto themselves. Please Google for some sources, because it's not possible to give more than a cursory overview here.
UPDATE: I recently became aware of the Microsoft Solver Foundation. I haven't looked into it myself, but perhaps it'd be helpful to C# developers in solving this problem.