Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids.In order to obtain a sufficiently stable higher order scheme with respect to the time and space coordinates,we develop a combination of the discontinuous Galerkin finite element(DGFE)method for the space discretization and the backward difference formulae(BDF)for the time discretization.Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step,we employ suitable linearizations of inviscid as well as viscous fluxes which give a linear algebraic problem at each time step.Finally,the resulting BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results are compared with reference data.

A SECOND ORDER UNCONDITIONALLY CONVERGENT FINITE ELEMENT METHOD FOR THE THERMAL EQUATION WITH JOULE HEATING PROBLEM

In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential,and establish optimal L^(2) error estimates for the fully discrete finite element solution without any restriction on the time-step size.The discrete solution is bounded in infinite norm.Finally,several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA

This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.

A Comparison of Semi-Lagrangian and Lagrange-Galerkin hp-FEM Methods in Convection-Diffusion Problems

We perform a comparison in terms of accuracy and CPU time between second order BDF semi-Lagrangian and Lagrange-Galerkin schemes in combination with high order finite element method.The numerical results show that for polynomials of degree 2 semi-Lagrangian schemes are faster than Lagrange-Galerkin schemes for the same number of degrees of freedom,however,for the same level of accuracy both methods are about the same in terms of CPU time.For polynomials of degree larger than 2,Lagrange-Galerkin schemes behave better than semi-Lagrangian schemes in terms of both accuracy and CPU time;specially,for polynomials of degree 8 or larger.Also,we have performed tests on the parallelization of these schemes and the speedup obtained is quasi-optimal even with more than 100 processors.