Reliable finite element methods for self-adjoint singular perturbation problems

It is well known that the standard finite element method based on the space Vh of continuous piecewise linear functions is not reliable in solving singular perturbation problems. It is also known that the solution of a two-point boundaryvalue singular perturbation problem admits a decomposition into a regular part and a finite linear combination of explicit singular functions. Taking into account this decomposition, first, we design a finite element method (which we call singular function method) where the space of trial and test functions is the direct sum of Vh and the space spanned by these singular functions. The second method, based on the finite element discretization on a suitably refined mesh, is referred to as mesh refinement method. Both of these methods are proved to be ε-uniformly convergent. Numerical examples which confirm the theory are presented.Quaestiones Mathematicae 32(2009), 397–41

Analysis of multilevel finite volume approximation of 2D convective Cahn–Hilliard equation

In this work, four finite volume methods have been constructed to solve the 2D convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. We prove existence and uniqueness of solutions. The stability and convergence analysis of the numerical methods have been discussed thoroughly. The nonlinear terms are approximated by a linear expression based on Mickens’ rule (Mickens, Nonstandard finite difference models of differential equations. World Scientific, Singapore, 1994) of nonlocal approximations of nonlinear terms. Numerical experiments for a test problem have been carried out to test all methods.http://link.springer.com/journal/131602018-04-30hb2017Mathematics and Applied Mathematic