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We consider the Helmholtz equation with a discontinuous complex parameter and inhomogeneous Dirichlet boundary conditions in a rectangular domain. A variant of the direct method of cyclic reduction (CR) is employed to facilitate the design of improved multigrid (MG) components, resulting in the method of CR-MG. We demonstrate the improved convergence properties of this method.
MgCr is Tetraauricupride structured and crystallizes in the tetragonal P4/mmm space group. The structure is three-dimensional. Mg is bonded in a body-centered cubic geometry to eight equivalent Cr atoms. All Mg–Cr bond lengths are 2.88 Å. Cr is bonded in a body-centered cubic geometry to eight equivalent Mg atoms.
Mg3Cr is beta Cu3Ti-like structured and crystallizes in the tetragonal I4/mmm space group. The structure is three-dimensional. there are two inequivalent Mg sites. In the first Mg site, Mg is bonded to eight equivalent Mg and four equivalent Cr atoms to form MgMg8Cr4 cuboctahedra that share corners with four equivalent MgMg8Cr4 cuboctahedra, corners with eight equivalent CrMg12 cuboctahedra, edges with twenty-four MgMg8Cr4 cuboctahedra, faces with six equivalent CrMg12 cuboctahedra, and faces with twelve MgMg8Cr4 cuboctahedra. All Mg–Mg bond lengths are 3.08 Å. All Mg–Cr bond lengths are 3.07 Å. In the second Mg site, Mg is bonded to eight Mg and four equivalent Cr atoms to form MgMg8Cr4 cuboctahedra that share corners with twelve equivalent MgMg8Cr4 cuboctahedra, edges with eight equivalent CrMg12 cuboctahedra, edges with sixteen MgMg8Cr4 cuboctahedra, faces with four equivalent CrMg12 cuboctahedra, and faces with fourteen MgMg8Cr4 cuboctahedra. All Mg–Mg bond lengths are 3.07 Å. All Mg–Cr bond lengths are 3.08 Å. Cr is bonded to twelve Mg atoms to form CrMg12 cuboctahedra that share corners with four equivalent CrMg12 cuboctahedra, corners with eight equivalent MgMg8Cr4 cuboctahedra, edges with eight equivalent CrMg12 cuboctahedra, edges with sixteen equivalent MgMg8Cr4 cuboctahedra, faces with four equivalent CrMg12 cuboctahedra, and faces with fourteen MgMg8Cr4 cuboctahedra.