On the convergence of Galerkin methods for auto-convolution Volterra integro-differential equations

The Galerkin method is proposed for initial value problem of auto-convolution Volterra integro-differential equation (AVIDE). The solvability of the Galerkin method is discussed, and the uniform boundedness of the numerical solution is provided by defining a discrete weighted exponential norm. In particular, it is proved that the quadrature Galerkin method obtained from the Galerkin method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the continuous piecewise polynomial collocation method. For the Galerkin approximated solution in continuous piecewise polynomial space of degree \(\varvec{m}\) $\mathit{m}$ , at first, the \(\varvec{m}\) $\mathit{m}$ global convergence order is obtained. By defining a projection operator, the convergence is improved, and the optimal \(\varvec{m+1}\) $\mathit{m}\mathbf{+}\mathbf{1}$ global convergence order is gained, as well as \(\varvec{2m}\) $\mathbf{2}\mathit{m}$ local convergence order at mesh points. Furthermore, all the above analysis for uniform mesh can be extended to typical quasi-uniform meshes. Some numerical experiments are given to illustrate the theoretical results.