On the robustness and optimality of algebraic multilevel methods for reaction–diffusion type problems

Abstract

This paper is on preconditioners for reactiondiffusion problems that are both, uniform with respect to the reaction-diffusion coefficients, and optimal in terms of computational complexity. The considered preconditioners belong to the class of so-called algebraic multilevel iteration (AMLI) methods, which are based on a multilevel block factorization and polynomial stabilization. The main focus of this work is on the construction and on the analysis of a hierarchical splitting of the conforming finite element space of piecewise linear functions that allows to meet the optimality conditions for the related AMLI preconditioner in case of second-order elliptic problems with non-vanishing zeroorder term. The finite element method (FEM) then leads to a system of linear equations with a system matrix that is a weighted sum of stiffness and mass matrices. Bounds for the constant γ in the strengthened Cauchy-BunyakowskiSchwarz inequality are computed for both mass and stiffness matrices in case of a general m-refinement. Moreover, an additive preconditioner is presented for the pivot blocks that arise in the course of the multilevel block factorization. Its optimality is proven for the case m = 3. Together with the Communicated by Gabriel Wittum. estimates for γ this shows that the construction of a uniformly convergent AMLI method with optimal complexity is possible (for m ≥ 3). Finally, we discuss the practical application of this preconditioning technique in the context of time-periodic parabolic optimal control problems.

Functional A Posteriori Error Estimates for Time-Periodic Parabolic Optimal Control Problems

“…Fast and robust solvers for the systems (19) and (20) can be found in [18,23,28,37], which we use in order to obtain the MhFE approximations…”

Classroom Visualization Based on the Relationship between User Roles and Scene Content

“…Such user-centered maps, also called personalized maps and adaptive maps, are important applications of indoor maps and location services. By establishing the mapping relationship between users and maps, the map expression content and representation method are dynamically changed in real time, and the personalized service method of the map is changed from the operation interaction between users and maps to autonomous push [7,8]. erefore, it is a very meaningful research direction to make full use of the visual advantages of 3D maps, to integrate user cognitive features and specific needs with map display interaction, to dynamically and effectively visualize the indoor scenes of classrooms, and to realize user personalized 3D visualization of indoor scenes.…”

Functional A Posteriori Error Estimates for Parabolic Time-Periodic Boundary Value Problems

“…In Table 1, we observe the robustness and optimality of the AMLI preconditioned MINRES method as presented in [17,29]. More precisely, the computational times increase with a factor of nine that exactly reveals the optimal computational complexity of the method according to the 3-refinement of the mesh.…”

“…We mention that, in all tables where the number of MINRES iterations n iter MINRES or of AMLI iterations n iter AMLI is presented, the iteration was stopped after reducing the initial residual by a factor of 10 −6 . In each MINRES iteration step, we have used the AMLI preconditioner according to [17] with 8 inner iterations. The presented CPU times in seconds t sec include the computational times for computing the majorants, which are very small in comparison to the computational times of the solver.…”