Connections between finite difference and finite element approximations

ABSTRACT

We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided.

KEYWORDS:

• Finite difference
• composite trapezoid rule
• finite element
• Green function
• dual vector
• degrees of freedom
• interpolation

2000 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65N06
• 65N12
• 65N22
• 65N30
• 65N80

Acknowledgments

The authors are grateful for an Oden Institute fellowship which allowed them to visit UT Austin in the Fall of 2019.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by NSF DMS [grant number 2011615].