• Abstract

This paper is devoted to a review of the prolate spheroidal wave functions (PSWFs) and their variants from the viewpoint of spectral ⁄ spectral-element approximations using such functions as basis functions. We demonstrate the pros and cons over their polynomial counterparts, and put the emphasis on the construction of essential building blocks for efficient spectral algorithms.

• Abstract

In this paper, we use Pacard-Xu's methods to discuss the complex deformation of constant scalar curvature metrics in the case of fixed and varying complex structures. Moreover, we also discuss the complex deformation of Kähler-Ricci solitons.

• Abstract

In this paper we use the maximum principle and the Hopf lemma to prove symmetry results to some overdetermined boundary value problems for domains in the hemisphere or star-shaped domains in $S^n$. Ourmethod is based on finding suitable P-functions as Weinberger ([26]).

• Abstract

Numerical simulations by high order methods for the blood flow model in arteries have wide applications in medical engineering. This blood flow model admits the steady state solutions, for which the flux gradient is non-zero, and is exactly balanced by the source term. In this paper,we design a high order discontinuous Galerkin method to this model by means of a novel source term approximation as well as wellbalanced numerical fluxes. Rigorous theoretical analysis as well as extensive numerical results all suggest that the resulting method maintains the well-balanced property, enjoys high order accuracy and keeps good resolutions for smooth and discontinuous solutions.

• Abstract

We prove the concavity of the power of a solution is preserved for a class of doubly nonlinear parabolic equation, which is a well-known feature in some particular cases such as the porous medium equation or the parabolic p-Laplace equation.