Optimal Computational Complexity Research Articles

An Adaptive Finite Element Method for Two-Phase Stefan Problems in Two Space Dimensions. II: Implementation and Numerical Experiments

An adaptive local mesh refinement strategy for two-phase Stefan problems is discussed in light of its efficiency and computational complexity. Three local parameters are used to equidistribute interpolation errors in maximum norm for temperature and a fourth one, in the event of mushy regions, to equidistribute $L^1 $-interpolation errors for enthalpy within the mush. If certain quality mesh tests fail, then the current mesh is discarded and a new one completely regenerated by an efficient mesh generator, which in turn is briefly described. A typical triangulation is strongly graded to become very fine near computed interfaces and coarse away from them. Consecutive meshes are not compatible. The use of quadtree data structures is discussed as a means to reach a nearly optimal computational complexity in various tasks to be performed, mainly in generating a mesh and interpolating. Various implementation details are given so as to derive the computational complexity of each relevant subroutine. The approximation of both solutions and interfaces is drastically improved. The proposed method is robust in that it can handle the formation of cusps and mushy regions as well as the spontaneous appearance of phases. It is also superior in terms of computing time or a desired accuracy. Several numerical experiments illustrate these facts and provide quantitative information about each task complexity.

Multigrid methods for two- and three-dimensional Poisson-type equations on the sphere

Abstract The efficient solution of two- and three-dimensional Poisson- and Helmholtz-type equations on the sphere with multigrid methods is the subject of this work. The strong anisotropic behavior of the Laplacian in spherical coordinates poses severe requirements on the relaxation procedures, in order to attain a full multigrid effectiveness. The introduction of a new technique, combined relaxation, results in high efficiency, even in cases with varying anisotropy direction, where alternating line or plane relaxations would normally be required. The combination of this new relaxation technique with semi-coarsening strategies is also studied. Even for moderate grid sizes, the multigrid solvers developed are shown to be competitive with two-dimensional, direct fast solvers, in cases where the latter can be applied. The optimal computational complexity of the multigrid methods, their broad applicability (in contrast to fast direct solvers), and their high efficiency, also in three spatial dimensions, make these methods very interesting for applications.