A Mixed Finite Element Discretisation of Linear and Nonlinear Multivariate Splines Using the Laplacian Penalty Based on Biorthogonal Systems

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.