Applied and Numerical Analysis Seminar - Numerical computation and analysis for time-dependent PDEs with non-smooth initial data or low-regularity solution

We mainly focus on numerical methods for nonlinear evolution equations with low-regularity solutions. The main academic achievements include: 1) Proposed a novel exponential integrator for solving nonlinear diffusion and subdiffusion equations with singular initial value problems, establishing optimal high-order error estimates and overcoming the first-order convergence bottleneck of traditional methods in time; 2) Introduced a time-variable step size technique to overcome the order reduction challenges faced by traditional methods when solving nonlinear Navier-Stokes with singular initial value problems; 3) Developed a low-regularity exponential integrator for solving nonlinear Navier-Stokes equations with small viscosity, significantly reducing the regularity requirements of solutions while achieving uniform convergence independent of the viscosity; 4) Developed an interface-tracking finite element method for solving two phase Navier-Stokes flow using high order curved evolving mesh. We rigorously proved the high-order convergence of this method in sharp interface models with low global regularity on the entire domain.