The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity proper...The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided.Then the existence and uniqueness of the DG solution are proven.Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established.Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants,the DG method achieves the uniform convergence in the L2 norm with respect to the singular perturbation parameter e when the space of polynomials with degree p is used.A numerical experiment validates the theoretical results.Furthermore,an ultra-convergence order 2p+1 at the nodes for the one-sided flux,uniform with respect to the singular perturbation parameter e,is observed numerically.展开更多
基金supported by the National Natural Science Foundation of China(12001189)supported by the National Natural Science Foundation of China(11171104,12171148)。
文摘The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided.Then the existence and uniqueness of the DG solution are proven.Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established.Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants,the DG method achieves the uniform convergence in the L2 norm with respect to the singular perturbation parameter e when the space of polynomials with degree p is used.A numerical experiment validates the theoretical results.Furthermore,an ultra-convergence order 2p+1 at the nodes for the one-sided flux,uniform with respect to the singular perturbation parameter e,is observed numerically.
