Conforming and nonconforming harmonic finite elements

ABSTRACT

Instead of using the full polynomial space, a conforming and a nonconforming finite element methods are designed where only harmonic polynomials (a much smaller space) are employed in the computation. The conforming quadratic harmonic polynomial finite element is defined only on a special triangular grid. The nonconforming quadratic harmonic finite element is defined on general triangular grids. The optimal order of convergence is proved for both finite element methods, and confirmed by numerical computations. In addition, numerical comparisons with the standard conforming and nonconforming finite elements are presented.

COMMUNICATED BY:

• Constantin Bacuta

KEYWORDS:

• Harmonic polynomial
• conforming finite element
• nonconforming finite element
• triangular grid
• Laplace equation

AMS SUBJECT CLASSIFICATIONS:

• 65N30
• 65N15

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Tatyana Sorokina  http://orcid.org/0000-0003-2486-1932

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

Additional information

Funding

Tatyana Sorokina is partially supported by the grant from the Simons Foundation [grant no 235411]. Shangyou Zhang is partially supported by The National Natural Science Foundation of China (NSFC) [grant no 11571023].