A monotone finite element method with test space of Legendre polynomials

Abstract

This paper is concerned with the development of a multi-dimensional monotone scheme to deal with erroneous oscillations in regions where sharp gradients exist. The strategy behind the underlying finite element analysis is the accommodation of the M-matrix to the Petrov-Galerkin finite element model. An irreducible diagonal-dominated coefficient matrix is rendered through the use of exponential weighting functions. With a priori knowledge capable of leading to a Monotone matrix, the analysis model is well conditioned with the monotonicity-preserving property. In order to stress the effectiveness of test functions in resolving oscillations, we considered two classes of the convection-diffusion problem. As seen from the computed results, we can classify the proposed finite element model as legitimate for the problem free of boundary layer. Also, through the use of this model, we can capture the solution for the problem involving a high gradient. In this study, we are interested in a cost-effective method which ensures monotonicity irrespective of the value of the Peclet number throughout the entire domain. To gain access to these desired properties, it is tempting to bring in the Legendre polynomials and the characteristic information so that by virtue of the inherent orthogonal property the integral can be obtained exactly by two Gaussian integration points along each spatial direction while maintaining stability in the M-matrix satisfaction sense.