Maxwell

Maxwell is a parallel solver for edge finite element discretization of the curl-curl formulation of the Maxwell equation

$\nabla \times \alpha \nabla \times E + \beta E= f, \beta> 0$

on semi-structured grids. Details of the algorithm can be found in [JoLe2006]. The solver can be viewed as an operator-dependent multiple-coarsening algorithm for the Helmholtz decomposition of the error correction. Input to this solver consist of only the linear system and a gradient operator. In fact, if the orientation of the edge elements conforms to a lexicographical ordering of the nodes of the grid, then the gradient operator can be generated with the routine HYPRE_MaxwellGrad: at grid points $$(i,j,k)$$ and $$(i-1,j,k),$$ the produced gradient operator takes values $$1$$ and $$-1$$ respectively, which is the correct gradient operator for the appropriate edge orientation. Since the gradient operator is normalized (i.e., $$h$$ independent) the edge finite element must also be normalized in the discretization.

This solver is currently developed for perfectly conducting boundary condition (Dirichlet). Hence, the rows and columns of the matrix that corresponding to the grid boundary must be set to the identity or zeroed off. This can be achieved with the routines HYPRE_SStructMaxwellPhysBdy and HYPRE_SStructMaxwellEliminateRowsCols. The former identifies the ranks of the rows that are located on the grid boundary, and the latter adjusts the boundary rows and cols. As usual, the rhs of the linear system must also be zeroed off at the boundary rows. This can be done using HYPRE_SStructMaxwellZeroVector.

With the adjusted linear system and a gradient operator, the user can form the Maxwell multigrid solver using several different edge interpolation schemes. For problems with smooth coefficients, the natural Nedelec interpolation operator can be used. This is formed by calling HYPRE_SStructMaxwellSetConstantCoef with the flag $$>0$$ before setting up the solver, otherwise the default edge interpolation is an operator-collapsing/element-agglomeration scheme. This is suitable for variable coefficients. Also, before setting up the solver, the user must pass the gradient operator, whether user or HYPRE_MaxwellGrad generated, with HYPRE_SStructMaxwellSetGrad. After these preliminary calls, the Maxwell solver can be setup by calling HYPRE_SStructMaxwellSetup.

There are two solver cycling schemes that can be used to solve the linear system. To describe these, one needs to consider the augmented system operator

$\begin{split}\bf{A}= \left [ \begin{array}{ll} A_{ee} & A_{en} \\ A_{ne} & A_{nn} \end{array} \right ],\end{split}$

where $$A_{ee}$$ is the stiffness matrix corresponding to the above curl-curl formulation, $$A_{nn}$$ is the nodal Poisson operator created by taking the Galerkin product of $$A_{ee}$$ and the gradient operator, and $$A_{ne}$$ and $$A_{en}$$ are the nodal-edge coupling operators (see [JoLe2006]). The algorithm for this Maxwell solver is based on forming a multigrid hierarchy to this augmented system using the block-diagonal interpolation operator

$\begin{split}\bf{P}= \left[ \begin{array}{ll} P_e & 0 \\ 0 & P_n \end{array} \right],\end{split}$

where $$P_e$$ and $$P_n$$ are respectively the edge and nodal interpolation operators determined individually from $$A_{ee}$$ and $$A_{nn}.$$ Taking a Galerkin product between $$\bf{A}$$ and $$\bf{P}$$ produces the next coarse augmented operator, which also has the nodal-edge coupling operators. Applying this procedure recursively produces nodal-edge coupling operators at all levels. Now, the first solver cycling scheme, HYPRE_SStructMaxwellSolve, keeps these coupling operators on all levels of the V-cycle. The second, cheaper scheme, HYPRE_SStructMaxwellSolve2, keeps the coupling operators only on the finest level, i.e., separate edge and nodal V-cycles that couple only on the finest level.