A free boundary problem for the chemotaxis model of parabolic-elliptic type is investigated in the present paper, which can be used to simulate the dynamics of cell density under the influence of the nonlinear diffusion and nonlocal attraction-repulsion forces. In particular, it is shown for supercritical case that if the initial total mass of cell density is small enough or the interaction between repulsion and attraction cancels almost each other, the strong solution for the cell density exists globally in time and converges to the self-similar Barenblatt solution at the algebraic time rate, and for subcritical case that if the initial data is a small perturbation of the steady-state solution and the attraction effect dominates the process, the strong solution for cell density exists globally in time and converges to the steady-state solution at the exponential time rate.

In this work, implicit and explicit multilevel finite volume methods have been constructed to solve the 2D Navier-Stokes equation with specified initial condition and boundary conditions. The multilevel methods are applied to the pressure-correction projection method using space finite volume discretization. The convective term is approximated by a linear expression that preserves the physical property of the continuous model. The stability analysis of the numerical methods have been discussed thoroughly by making use of the energy method. Numerical experiments exhibited to illustrate some differences between the new (multilevel) and conventional (one-level) schemes.

The weak Galerkin (WG) finite element method is an effective and robust numerical technique for the approximate solution of partial differential equations. The essence of the method is the use of weak finite element functions and their weak derivatives computed with a framework that mimics the distribution or generalized functions. Weak functions and their weak derivatives can be constructed by using polynomials of arbitrary degrees; each chosen combination of polynomial subspaces generates a particular set of weak Galerkin finite elements in application to PDE solving. This article explores the computational performance of various weak Galerkin finite elements in terms of stability, convergence, and supercloseness when applied to the model Dirichlet boundary value problem for a second order elliptic equation. The numerical results are illustrated in 31 tables, which serve two purposes: (1) they provide detailed and specific guidance on the numerical performance of a large class of WG elements, and (2) the information shown in the tables may open new research projects for interested researchers as they interpret the results from their own perspectives.

Error estimates of finite element methods for reaction-diffusion problems are often realised in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the $H$^$m$ seminorm for 2$m$-th order problems leads to a balanced norm which reflects the layer behaviour correctly. We prove error estimates in such balanced norms and improve thereby existing estimates known in literature.

The new time high-order energy-preserving schemes are proposed for the nonlocal Benjamin-Ono equation. We get the Hamiltonian system to the nonlocal model, and it is then discretized by a Fourier pseudospectral method in space and the Hamiltonian boundary value method (HBVM) in time. This approach has high order of convergence in time and conserves the total mass and energy in discrete forms. We further develop a time second-order energy-preserving scheme and a time fourth-order energy-preserving scheme for the nonlocal Benjamin-Ono equation. Numerical experiments test the proposed schemes with a single solitary wave and the interaction of two solitary waves. Results confirm the accuracy and conservation properties of the schemes.

We investigate the transient dynamics of Block Coordinate Descent algorithms in valleys of the optimization landscape. Iterates converge linearly to a vicinity of the valley floor and then progress in a zig-zag fashion along the direction of the valley floor. When the valley sides are symmetric, the contraction factor to a vicinity of the valley floor appears to be no worse than 1/8, but without symmetry the contraction factor can approach 1. Progress along the direction of the valley floor is proportional to the gradient on the valley floor and inversely proportional to the "narrowness" of the valley. We quantify narrowness using the eigenvalues of the Hessian on the valley floor and give explicit formulas for certain cases. Progress also depends on the direction of the valley with respect to the blocks of coordinates. When the valley sides are symmetric, we give an explicit formula for this dependence and use it to show that in higher dimensions nearly all directions give progress similar to the worst case direction. Finally, we observe that when starting the algorithm, the ordering of blocks in the first few steps can be important, but show that a greedy strategy with respect to objective function improvement can be a bad choice.

In this paper, we study the rate of convergence of a sequential quadratic programming (SQP) method for nonlinear semidefinite programming (SDP) problems. Since the linear SDP constraints does not contribute to the Hessian of the Lagrangian, we propose a reduced SQP-type method, which solves an equivalent and reduced type of the nonlinear SDP problem near the optimal point. For the reduced SDP problem, the well-known and often mentioned "$σ$-term" in the second order sufficient condition vanishes. We analyze the rate of local convergence of the reduced SQP-type method and give a sufficient and necessary condition for its superlinear convergence. Furthermore, we give a sufficient and necessary condition for superlinear convergence of the SQP-type method under the nondegeneracy condition, the second-order sufficient condition with $σ$-term and the strict complementarity condition.