Analysis and computation of corner singularity for some parabolic and hyperbolic equations

In this dissertation, the corner singularities for some equations, namely, nonlinear diffusive equations, heat equations and the Korteweg de Vries (KdV) equations are considered.;For 1D nonlinear diffusive equations, singular corner functions are introduced in conjunction with the Galerkin finite element method. In this dissertation, this procedure is called the correction procedure. The numerical results confirm the effectiveness of the proposed procedure.;For higher dimensional heat equations, the construction of the singular corner functions remains an open problem, so a new method, based on the so-called penalty method, is proposed to improve the numerical simulations. On the one hand, we prove the strong convergence of the solutions of the approximated (penalized) system to the solutions of the original system; on the other hand, the numerical results confirm the effectiveness of the new method, and furthermore this penalty based method solves a very important issue caused by the corner singularities: no matter how much we refine our mesh, the errors at initial time steps do not decrease otherwise. This is the first time that a solution to the corner singularity problem is ever proposed in dimension higher than one.;For the 1D KdV equations, we also apply the penalty method to handle the incompatibility issue. We theoretically prove the convergence of the solutions of the approximated system to the solutions of the original system, while the numerical tests confirm the effectiveness of the method. Note that the equation being now hyperbolic, the numerical difficulties are even more important, as the corner singularities propagate in the whole domain through the characteristics.