Adaptive multigrid methods are described that can be used to solve large-scale structural mechanics problems on parallel computers using the finite element method. The multigrid method was efficiently implemented on shared memory parallel computers by noting that the algorithm is dominated by matrix-vector products, and then performing these operations on the element level. Not only does this approach produce efficient code, but the storage requirements are reduced, since no element stiffness matrices need to be assembled and stored. This requires the development of a method for performing element level matrix-vector multiplications for h-refined meshes. The solution of a simple test problem demonstrates the performance of the multigrid method. It was observed that a single multigrid cycle is sufficient to obtain an approximate solution to within the discretization error of the mesh being used.

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