Algebraic Hybridization and Static Condensation with Application to Scalable $H$(div) Preconditioning

Abstract

We propose an unified algebraic approach for static condensation and hybridization, two popular techniques in finite element discretizations. The algebraic approach is supported by the construction of scalable solvers for problems involving H(div)-spaces discretized by conforming (Raviart-Thomas) elements of arbitrary order. We illustrate through numerical experiments the relative performance of the two (in some sense dual) techniques in comparison with a state-of-the-art parallel solver, ADS [24], available in [1] and [2]. The superior performance of the hybridization technique is clearly demonstrated with increased benefit for higher order elements.1. Introduction. This paper proposes a unified algebraic approach of constructing reduced problems by two popular techniques in the finite element method: static condensation and hybridization. It also discusses approaches for constructing scalable preconditioners for these reduced problems. The present work is motivated by the solution of problems arising in variety of finite element simulations and of particular interest is the case when the problems involve the function space H(div) (L 2 -vector functions that have their weak divergence in L 2 ). Such problems naturally arise in the mixed finite element method for Darcy equation [6], Brinkman equation [27], as well as in the FOSLS (first order system least-squares) discretization approach, [11]. Also, certain formulations of radiation-diffusion transport [9], lead to problems involving the space H(div). This is in fact one of the practical examples from a more realistic simulation that we consider in our numerical tests.Highly scalable solvers for H(div) problems have been designed in the past. The most successful ones, of ADS-type [24], were based on the regular decomposition theory developed by Hiptmair and Xu [20]. Although of optimal complexity, the current state-of-the-art ADS algorithms are quite involved and require also going through solvers for H(curl), such as AMS, [23]. Both solvers, ADS and AMS, require some additional user input (e.g., discrete gradient), which makes them not completely algebraic. The results of the present paper can be viewed as a further step towards the design of fully algebraic solvers for problems involving H(div).To design faster, yet still scalable, alternative to ADS, we employ a traditional finite element technique commonly used in mixed finite elements referred to as hybridization. This is perhaps the first technique proposed to solve the saddle-point problems arising in the mixed methods, since it leads to symmetric and positive definite (SPD) reduced problems. Its early form appeared in a paper by Fraeijis de Veubekein in 1965 [17] as an efficient solution strategy for elasticity problems. In [3], Arnold and Brezzi studied the method from a theoretical perspective, and obtained *

Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation

“…To this end, we again make use of the availability of local matrices (during the coarsening steps) and transform the saddle point problems into symmetric positive-definite problems via hybridization, which is a classic technique for solving discrete problems arising from mixed finite element discretization of partial differential equations [5]. Recently in [26], and [12], an algebraic multigrid with some proper diagonal rescaling is demonstrated be a competitive solver for the hybridized system. Because the solver is constructed algebraically, it can be applied to solve saddle point problems that are not coming from finite element discretizations, which is the case in the current paper.…”

An observation on the uniform preconditioners for the mixed Darcy problem

“…Second, we remark that the arguments below are similar to [12], although the Sobolev spaces and bilinear forms involved are different. We now verify all of the requisite Brezzi conditions for (13) and (14). Towards this end let…”

“…Proposition 1 (A uniform-in-K continuous stability result). Consider the problem (13) and (14) where u and p are chosen, respectively, in the spaces:…”

“…provides a robust preconditioner for (13) and (14) in the sense that the condition number of  is bounded uniformly in K. Here,  is given by (15).…”

“…Proof Preconditioners for the V inner product are well‐known, cf. for example, [4, 13–16]. Following the framework of [1] it therefore suffices to construct an operator realizing the Q ‐norm.…”