Which is the best approach to solve a PDE in parallel:
1.To split the mesh the mesh in N parts and every processor works on its own part or
2.To take the global linear system Ax=b and solve it in parallel
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At some point, even processors working on their own of course have to communicate their work somehow. Your approach 1 then naturally leads to a class of methods that are called "Domain Decomposition" (DD) where each processor (repeatedly) solves problems that correspond only to the cells it "owns".
The second approach of course also has to split the mesh somehow because the linear system $Ax=b$ does not magically appear out of thin air, but needs to be constructed and that is also done by partitioning the mesh.
My take is that over the past 20 years, we've learned that approach 2 is the way to go. DD methods had a good run in the late 1990s and early 2000s, but when software came around that allowed for efficient handling and solving of globally distributed linear systems, that approach won out (for what I think are very good reasons). So go with the second method.