Numerical Methods For Singularly Perturbed And Interface Problems

In many science and engineering fields,we need to solve initial boundary value problems of partial differential equations with singularities by numerical methods.Many mathematicians and engineers have paid attention to these problems.In this dissertation,we discuss numerical methods for parabolic singularly perturbed problems with small parameters and second order elliptic interface problems in star-shaped domains.We first propose the tailored finite point method for parabolic singularly perturbed problems.Because of the small parameter in the highest order term,the solution of the problem often contains boundary layers or interior layers.In boundary/interior layers,the solution changes rapidly,so it is very difficult to capture this kind of change.The main idea of our tailored finite point method is using the solution of local reduced equation as the basis function to construct numerical schemes of the original problem.In this dissertation,we first consider a simple heat conduction equation.By using heat polynomials as basis functions,we can recover traditional finite difference schemes.Then we design numerical schemes for reaction-diffusion-convection problems.By coupling our scheme with Shishkin mesh,we get a uniformly convergent method,and the ?-uniform maximum norm error estimate can be obtained.Numerical results demonstrate the efficiency of our method.Then we discuss the direct method of lines for second order elliptic interface problem in a star-shaped domain.Firstly,by using a transformation of coordinates,we can reduce the original problem to a boundary value problem with discontinuous coefficients on a semi-infinite strip domain.Then the semi-discrete approximation of the new problem is obtained by applying finite element method.So the problem is equivalent to a boundary value problem of a system of ordinary differential equations.Solving this problem directly,we obtain the semi-discrete approximation of the original problem finally.The error estimate is also obtained.And numerical results show that we can achieve a high accuracy without any information of the singularities of the problem.