The Singularity-Seperated Method For Solving Singularly Perturbed Problem

Singularly perturbed problem is common in scientific research and engineering application.Because of the small parameter ?>0,the solution of the differential equation will change acutely in a narrow region,so it is difficult to be solved.Oscillation will occur in the numerical solution when the singularly perturbed problem is solved by traditional numerical method.The local refinement meshes is effective for the computation of singularly perturbed problem,but there are some shortcomings when they are used in multi-dimensional problem.The large aspect ratio of some grids will result in instability in numerical computation,and the computation load will be high.It is significant that how to keep a balance between the accuracy and computational cost.A new method named Singularity-Separated method is proposed innovatively in this paper.The main work and innovations are as follows:(1)The Singularity-Separated method is proposed for the first time to solve the singularly perturbed problems.The main idea is as follows:Firstly,a third boundary value auxiliary problem is constructed by modifying the boundary value condition in the original problem.The high-accuracy numerical solution w of this auxiliary problem can be gotten because of the weak singularity.Subsequently,a singularity function is constructed by the value of w in the boundary and the eigenvalue of differential equation.It can separate the singularity in the solution u of the original problem.It is notably that the singular function v doesn't need to be solved by numerical method.Finally,the solution u of the original problem can be expressed by w-v.A special mesh”N0+N1 meshes”is used in the numerical experiments.N0 is the number of the grids outside the boundary layer and N1 is the number of grids in the boundary layer.Let N1=1 in the numerical experiments and numerical solution with high accuracy can be gotten.It illustrates the superiority of Singularity-Separated method.(2)Singularity-Separated method is used to solve the one dimensional singularly perturbed two-point boundary value problem.For the third boundary value auxiliary problem,the energy estimates of w(x)are proved,especially the most important regularity estimate ?wxx??C which is uniformly bounded.The convergence of LDG solution are proved in this paper.Three finite element methods are used to implement the singularity-separated method in the numerical experiments.High accuracy numerical solutions are obtained under a few grids or uniform grids.The singularity-separated method is effective both in computation cost and high accuracy.(3)Singularity-Separated method is also used to solve the parabolic singularly perturbed convection-diffusion problem.The regularity estimates ?wx??+|||wx||||?C and |||wxx|||?C?-1/2 are derived.The high accuracy numerical solutions are also obtained by LDG method under a few grids,and the 2p+1 order uniform superconvergence of numerical fluxes are observed numerically.