A Hp-type C~0-continuous Petrov-Galerkin Method For Second-order Weakly Singular Volterra Integro-differential Equations

Volterra integro-differential equation(VIDE)arises from physics,chemistry,finance,biology and many other disciplines,and it has been widely used in many fields of scientific and engineering computation.In recent decades,the research on numerical methods of VIDEs has made great progress.However,most of the studies focus on the first-order VIDEs,and few on the numerical methods of the second-order or higher-order VIDEs.In this thesis,we consider high accuracy numerical methods for second-order VIDEs with weakly singular kernels.We develop an hp-version of the C~0-continuous Petrov-Galerkin(C~0-CPG)method for a class of second-order linear VIDEs with weakly singular kernels by combining the idea of continuous and discontinuous Galerkin methods.This method can discretize the second derivative of time directly and avoid the disadvantage of doubling the computational cost caused by the traditional reduced-order method.We prove the existence and uniqueness of the numerical solution provided that the time step is suitable small,and we also analyzed the convergence of the proposed method under arbitrary partitions.Moreover,for solutions with singularities at one of the endpoints,we prove exponential convergence of the hp-version C~0-continuous Petrov-Galerkin method based on geometric partitions.Finally,we verify the theoretical results by some numerical examples.