Numerical methods and analysis for some partial differential equations with nonsmooth structure

Abstract

Several kinds of partial differential equations with nonsmooth structure are considered in this thesis. Both numerical schemes and convergence analysis are presented for each kind of equation. Dynamic Ginzburg-Landau equation is considered over a possibly nonconvex and multi-connected (with holes) domain. The dynamic Ginzburg-Landau equation is reformulated into a new system of equations and the quantity B : ▽ x A is introduced as an unknown solution. Global well-posedness of the new system and its equivalence to the original problem are proved. A linearized and decoupled Galerkin finite element method is proposed for solving the new system. The convergence of numerical solutions is proved based on a compactness argument by utilizing the maximal Lp-regularity of the discretized equations. Time discretization of the time-dependent Stokes-Darcy system is considered with the k-step backward difference formula. The equation is converted to an abstract linear parabolic equation with a non-symmetric operator. The k-step backward difference formula is applied to the abstract parabolic equation, with implicit scheme for the symmetric part and the explicit k-step extrapolation scheme for the anti-symmetric part. An initial correction scheme is proposed for compensating the lack of k-step extrapolation at the starting k - 1 steps and improving the accuracy to optimal order. Long-time kth-order convergence for the numerical method is proved for initial data in H1Ω. For parabolic interface problem with general L2 initial value, a numerical approximation scheme is proposed. The problem is discretized by the finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete solution is further discretized by k-step backward difference formula. An error bound of O(t-knTk + t-1nh2|logh|) is nn established for the fully discrete finite element method. A high-order time stepping schemes is developed for semilinear subdiffusion equations. Due to the nonlinearity of the source term, only optimal order convergence rate can be achieved by the corrected backward difference formula, whose convergence order is proved to be O(Tmin(k,1 + 2α - ε)). The analysis is derived with the help of splitting the nonlinear potential term into an irregular linear part and a more regular nonlinear part and of the generating function technique, without further assumptions required on the regularity of the solution. Numerical examples are provided to support the theoretical results.