A Multigrid Method for Nonlinear Unstructured Finite Element Elliptic Equations

This paper presents an application of the element agglomeration-based coarsening procedure (agglomeration AMGe) proposed in [10], to build the components of a multigrid method for solving nonlinear finite element elliptic equations on general unstructured meshes. The agglomeration-based AMGe offers the ability to define coarse elements and element matrices, provided access to elements and element matrices on the fine grid is available. We focus on the performance of the classical full approximation scheme (FAS). In the present context the coarse nodes are constructed algebraically based on the element agglomeration, and the interpolation rules are based on the (linear) AMGe exploiting element matrices of Laplace operator and L2-mass element matrices. The AMGe provides the coarse counterparts on all levels. The nonlinear coefficients are averaged over the coarse elements, which leads to non-inherited forms and hence to non-inherited multigrid methods. Numerical results show that the resulting nonlinear multigrid gives mesh independent convergence on model problems. In addition, the nonlinear multigrid scheme appears to be more efficient and robust for poor initial guesses than repeated applications of the nonlinear system smoother (i.e., single level method). Finally, our numerical results indicate that handling nonlinearities on coarse grids can provide an advantage over nonlinear solvers that handle nonlinearities only on the original problem grid.