A discontinuous least-squares finite-element method for second-order elliptic equations

ABSTRACT

In this paper, a discontinuous least-squares (DLS) finite-element method is introduced. The novelty of this work is twofold, to develop a DLS formulation that works for general polytopal meshes and to provide rigorous error analysis for it. This new method provides accurate approximations for both the primal and the flux variables. We obtain optimal-order error estimates for both the primal and the flux variables. Numerical examples are tested for polynomials up to degree 4 on non-triangular meshes, i.e. on rectangular and hexagonal meshes.

KEYWORDS:

• Discontinuous Galerkin
• finite-element methods
• least-squares finite-element methods
• second-order elliptic problems
• polygonal mesh

2010 MSC SUBJECT CLASSIFICATIONS:

• Primary: 65N15
• 65N30
• 76D07
• Secondary: 35B45
• 35J50

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Xiu Ye http://orcid.org/0000-0002-1559-3836

Shangyou Zhang http://orcid.org/0000-0002-1114-4179

Additional information

Funding

The research of the first author was supported in part by National Science Foundation Division of Mathematical Sciences (Grant no. DMS-1620016). The research of the second author was supported by National Natural Science Foundation of China (NSFC) (Grant no. 11571023).