Least squares preconditioning for mixed methods with nonconforming trial spaces

ABSTRACT

We consider a preconditioning technique for mixed methods with a conforming test space and a nonconforming trial space. Our method is based on the classical saddle point disccretization theory for mixed methods and the theory of preconditioning symmetric positive definite operators. Efficient iterative processes for solving the discrete mixed formulations are proposed and choices for discrete compatible spaces are provided. For discretization, a basis is needed only for the test spaces and assembly of a global saddle point system is avoided. We provide approximation properties for the discretization and iteration errors and also provide a sharp estimate for the convergence rate of the proposed algorithm in terms of the condition number of the elliptic preconditioner and the discrete $inf-sup$ and $sup-sup$ constants of the pair of discrete spaces. We focus on applications to elliptic PDEs with discontinuous coefficients. Numerical results for two- and three-dimensional domains are included to support the proposed method.

COMMUNICATED BY:

• Robert Gilbert

Keywords:

• Least squares
• saddle point systems
• mixed methods
• multilevel methods
• Uzawa type algorithms
• conjugate gradient
• cascadic algorithm
• dual DPG

2010 Mathematics Subject Classifications:

• 74S05
• 74B05
• 65N22
• 65N55

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by NSF, DMS-1522454.