libMesh

If you are looking for a highly-efficient library for numerical simulation of partial differential equations, the libMesh library is here to help you. It provides one of the best frameworks for simulation using arbitrary unstructured discretizations on both parallel and serial platforms. Whether you need to do 1D, 2D, or 3D steady and transient finite element simulations, libMesh has got you covered.
But it doesn't stop there! This library also makes use of other high-quality software, including PETSc for solving linear systems on both serial and parallel machines, and LASPack for linear solver support on serial platforms. In addition to this, it also provides an optional interface to SLEPc, which facilitates solving both standard and generalized eigenvalue problems.
With all these features, libMesh is an excellent choice for anyone looking for a solid library for numerical simulation of partial differential equations.