A Dirichlet boundary value problem for a linear parabolic differential equation is studied on a rectangular domain in the x− t plane. The coefficient of the second order space derivative is a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. It is proved that a numerical method, comprising a standard finite difference...

A class of singularly perturbed quasilinear differential equations with discontinuous data is examined. In general, interior layers will appear in the solutions of problems from this class. A numerical method is constructed for this problem class, which involves an appropriate piecewise-uniform mesh. The method is shown to be a parameter-uniform numerical method with respect to the singular per...

A singularly perturbed convection-diffusion problem, with a discontinuous convection coefficient and a singular perturbation parameter ε, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this problem, which generates ε-uniformly convergent numerical approximations to the solution. The method uses a piecewise u...

We consider the numerical approximation of singularly perturbed reaction-diffusion problems over twodimensional domains with smooth boundary. Using the h version of the finite element method over appropriately designed piecewise uniform (Shishkin) meshes, we are able to uniformly approximate the solution at a quasi-optimal rate. The results of numerical computations showing agreement with the a...

it is difficult to develop an algorithm which is able to generate the appropriate mesh around the interfaces in bimaterials. in this study, a corresponding algorithm is proposed for this class of unified structures made from different materials with arbitrary shapes. the non-uniform mesh is generated adaptively based on advancing front technique available in abaqus software. implementing severa...

In this paper we describe an experimental technique for computing realistic values of the parameter–uniform order of convergence and error constant in the maximum norm associated with a parameter–uniform numerical method for solving singularly perturbed problems. We employ the technique to compute Reynolds– uniform error bounds in the maximum norm for the numerical solutions generated by a fitt...

We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular (but possibly nonquasi-uniform) meshes. These inequalities involve norms of the form �h � u� W s,p (Ω) for positive and negative s and �, where h is a function which reflects the local mesh diameter in an appropriate way. Th...