Numerical analysis and computation of some nonlinear evolution equations

Abstract

This thesis is devoted to new computational methods and analysis for some nonlinear evolution equations. In Chapter 3 we propose a family of arbitrarily high-order fully discrete space-time finite element methods for the nonlinear Schrodinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and conserving both mass and energy at the discrete level. An error bound of the form 0(hp+τk+1) in the L∞(0, T;H1)-norm is established, where h and τ denote the spatial and temporal mesh sizes, respectively, and (p, k) is the degree of the space-time finite elements. Chapters 4 and 5 are devoted to the numerical solutions of semilinear parabolic and subdiffusion equations with nonsmooth initial data. In Chapter 4 we propose a multistep exponential integrator for the semilinear parabolic equation with initial data in L∞(Ω), with variable stepsizes to resolve the singularity of the solutions at t = 0 and with contour integral approximations to the operator-valued exponential functions. In Chapter 5 we extend this idea to the semilinear subdiffusion equation with initial data in L∞(Ω). We propose an exponential convolution quadrature which combines contour integral representation of the solution, quadrature approximation of contour integrals, multistep exponential integrators for ordinary differential equations, and locally refined stepsizes to resolve the initial singularity. The proposed k-step exponential integrator and exponential convolution quadrature can have kth-order convergence for bounded measurable solutions of the semilinear parabolic and subdiffusion equations, respectively, based on the natural regularity of the solutions corresponding to the bounded measurable initial data. Chapters 6 and 7 are concerned with the Navier–Stokes (NS) equations with nonsmooth H1 and L2 initial data, respectively. By utilizing special locally refined temporal stepsizes, in Chapter 6 we prove that the linearly extrapolated Crank–Nicolson finite element methods can achieve second-order convergence in time and space for the NS equations with H1 initial data. In Chapter 7, we prove first-order convergence in time and space for a fully discrete semi-implicit Euler finite element methods for the NS equations with L2 initial data, without extra grid-ratio conditions. The proof utilises the smoothing properties of the NS equations and an appropriate duality argument with variable temporal stepsizes to resolve the initial singularity in the consistency errors. In Chapter 8 we propose a semi-implicit exponential low-regularity integrator for the NS equations. The proposed method is proved to preserve the energy-decay structure of the NS equations. First-order convergence of the proposed method is established independently of the viscosity coefficient u, under weaker regularity conditions than other existing numerical methods, including the semi-implicit Euler method and classical exponential integrators. The proposed low-regularity integrator is extended to two types of full discretizations with a stabilized finite element method and a spectral collocation method, respectively. An error estimate of full discretization is established for the stabilized finite element method.