A new space–time continuous Galerkin method with mesh modification for Sobolev equations

In this study, we first propose the continuous Galerkin (CG) method for the Sobolev equations, which allows different temporal step-sizes and spatial grids in each time step. And then, we demonstrate the existence and uniqueness of the approximate solutions and derive the optimal rates of convergence of the approximate solutions under the restrictive assumptions that the space–time finite element subspaces between two successive time steps are conforming elements. Finally, we provide some numerical examples on unstructured meshes to demonstrate the efficiency and flexibility of this method.