The title is incorrect. It's not a general partial differential equation solver. The paper doesn't even claim it to be. The issue is that not all, not even most

The title is incorrect. It's not a general partial differential equation solver. The paper doesn't even claim it to be. The issue is that not all, not even most, PDEs can be discretized to a linear system of equations. Normally these need to be discretized to either a nonlinear rootfinding problem G(x)=0 find x, or a stiff ODE (normally solved by an implicit method, but not necessarily). However, both of those problems do include a subproblem which is solving linear systems (when the ODE is solved implicitly), and so PDE solvers which take these routes will generally be able to be accelerated by an improved linear equation solver. Thus it's not an be all and end all algorithm, but when combined with the tools that exist for PDEs in something like PETSc or Julia then this can be used to give a more efficient algorithm for most PDEs (though there are still other PDEs where linear solving is entirely avoided, for example with exponential integrators or Runge-Kutta Chebyshev methods, which can be more efficient depending on the problem).