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We present a comparison of different multigrid approaches for the solution ofsystems arising from high-order continuous finite element discretizations ofelliptic partial differential equations on complex geometries. We consider thepointwise Jacobi, the Chebyshev-accelerated Jacobi and the symmetric successiveover-relaxation (SSOR) smoothers, as well as elementwise block Jacobismoothing. Three approaches for the multigrid hierarchy are compared: 1)high-order $h$-multigrid, which uses high-order interpolation and restrictionbetween geometrically coarsened meshes; 2) $p$-multigrid, in which thepolynomial order is reduced while the mesh remains unchanged, and theinterpolation and restriction incorporate the different-order basis functions;and 3), a first-order approximation multigrid preconditioner constructed usingthe nodes of the high-order discretization. This latter approach is oftencombined with algebraic multigrid for the low-order operator and is attractivefor high-order discretizations on unstructured meshes, where geometriccoarsening is difficult. Based on a simple performance model, we compare thecomputational cost of the different approaches. Using scalar test problems intwo and three dimensions with constant and varying coefficients, we compare theperformance of the different multigrid approaches for polynomial orders up to16. Overall, both $h$- and $p$-multigrid work well; the first-orderapproximation is less efficient. For constant coefficients, all smoothers workwell. For variable coefficients, Chebyshev and SSOR smoothing outperformsJacobi smoothing. While all of the tested methods converge in amesh-independent number of iterations, none of them behaves completelyindependent of the polynomial order. When multigrid is used as a preconditionerin a Krylov method, the iteration number decreases significantly compared tousing multigrid as a solver.

Comparison of Multigrid Algorithms for High-order Continuous Finite Element Discretizations