A mixed finite element discretisation of linear and nonlinear multivariate splines using the Laplacian penalty based on biorthogonal systems

Bishnu P Lamichhane

doi:10.1016/j.mex.2022.101962

A mixed finite element discretisation of linear and nonlinear multivariate splines using the Laplacian penalty based on biorthogonal systems

MethodsX. 2022 Dec 13:10:101962. doi: 10.1016/j.mex.2022.101962. eCollection 2023.

Author

Bishnu P Lamichhane¹

Affiliation

¹ School of Information & Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW 2308, Australia.

Abstract

We consider a mixed finite element method for a linear multivariate spline using the Laplacian penalty. Our discretisation is based on biorthogonal systems leading to a very simple and efficient finite element scheme. We also extend our approach to a nonlinear case and describe a split Bregman iteration scheme for the resulting nonlinear equations. We apply our numerical schemes to remove the mixture of Gaussian and impulsive noise for some test images.•This paper presents a method of discretising a multivariate spline using a finite element method.•The method uses a biorthogonal system to achieve an efficient finite element method.•The method is extended to cover a discretisation scheme for a nonlinear case, including an adaptation of the split Bregman method for the nonlinear case.

Keywords: 41A15; 65D15; A finite element approximation of a multivariate spline model; Biorthogonal system; MSC:; Mixed finite element method; Multivariate spline; Scattered data smoothing; Split Bregman method; Total variation model.