The uniform convergence of a DG method for a singularly perturbed Volterra integro-differential equation. (English) Zbl 1515.65335
Summary: 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 \(L^2\) norm with respect to the singular perturbation parameter \(\epsilon\) 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 \(\epsilon\), is observed numerically.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45J05 | Integro-ordinary differential equations |
| 45D05 | Volterra integral equations |
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