Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems
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Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems Na Huang1 Received: 12 September 2018 / Accepted: 4 December 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this work, we consider numerical methods for solving a class of block three-bythree saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and a class of quadratic programs. We present a variant of Uzawa method with two variable parameters for the saddle point problems. These two parameters can be updated easily in each iteration, similar to the evaluation of the two iteration parameters in the conjugate gradient method. We show that the new iterative method converges to the unique solution of the saddle point problems under a reasonable condition. Numerical experiments highlighting the performance of the proposed method for problems are presented. Keywords Saddle point problem · Uzawa iterative method · Variable parameter Mathematics Subject Classification (2010) 65F10 · 65F50
1 Introduction In this work, we consider the following block three-by-three saddle point problems: ⎛ ⎞⎛ ⎞ ⎛ ⎞ A BT 0 f x A ≡ ⎝ B 0 C T ⎠ ⎝ y ⎠ = ⎝ g ⎠ , (1.1) h z 0 C 0 where A ∈ Rn×n is a symmetric positive definite matrix, B ∈ Rm×n and C ∈ R×m have full row rank, and f ∈ Rn , g ∈ Rm , and h ∈ R are given vectors. It is not
Na Huang
[email protected] 1
Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, China
Numerical Algorithms
difficult to verify that the coefficient matrix A is nonsingular, then the system (1.1) has an unique solution. Linear systems of form (1.1) have two important applications: •
The finite element methods for solving time-dependent Maxwell equations with discontinuous coefficients in general three-dimensional Lipschitz polyhedral domains [18]: ε∂t E − curl H μ∂t H + curl E div(ε E) div(μ H)
•
= = = =
J in × (0, T ), 0 in × (0, T ), ρ in × (0, T ), 0 in × (0, T ),
(1.2)
where ⊂ R3 is a simply connected Lipschitz polyhedral domain with connected boundary which is occupied by the dielectric material, E and H are the electric and magnetic fields, J and ρ are the current density and charge density, and ε and μ are the permittivity parameter and permeability parameter, which are discontinuous across an interface ⊂ . Here is the boundary of a simply ¯ 1 ⊂ and 2 = \ ¯ 1. connected Lipschitz polyhedral domain 1 with Karush-Kuhn-Tucker necessary conditions [7] of a class of quadratic programs [35]: 1 T T T min x Ax − f x − h z , s.t. Bx +C T z = g, x ∈ Rn , z ∈ R . (1.3) 2
Numerical solutions for linear systems play an important role in many applications, like diffuse optical tomography, computational fluid dynamics, lattice quantum chromodynamics, molecular scattering, electronic networks, computer graphics, and optimal control, see, e.g., [3, 8, 17, 22, 24, 46]. At present, quite many studies can be found in literature on the iterative methods
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