Parameter uniform convergence of a finite element method for a singularly perturbed linear reaction diffusion system with discontinuous source terms

A linear system of ’n’ second order ordinary differential equations of reaction-diffusion type with discontinuous source terms is considered. On a piecewise uniform Shishkin mesh, a numerical system is built that employs the finite element method. The numerical approximations obtained by this approach are proven to be effectively almost second order convergent

Second order parameter-uniform numerical method for a partially singularly perturbed linear system of reaction-diusion type

A partially singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading terms of first $m$ equations are multiplied by small positive singular perturbation parameters which are assumed to be distinct. The rest of the equations are not singularly perturbed. The first $m$ components of the solution exhibit overlapping layers and the remaining $n-m$ components have less-severe overlapping layers. Shishkin piecewise-uniform meshes are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximation obtained by this method is essentially second order convergent uniformly with respect to all the parameters. Numerical illustrations are presented in support of the theory

Second order parameter-uniform numerical method for a partially singularly perturbed linear system of reaction-diusion type

A partially singularly perturbed linear system of second order ordinary differential equations of reaction-diffusion type with given boundary conditions is considered. The leading terms of first $m$ equations are multiplied by small positive singular perturbation parameters which are assumed to be distinct. The rest of the equations are not singularly perturbed. The first $m$ components of the solution exhibit overlapping layers and the remaining $n-m$ components have less-severe overlapping layers. Shishkin piecewise-uniform meshes are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximation obtained by this method is essentially second order convergent uniformly with respect to all the parameters. Numerical illustrations are presented in support of the theory