在本文中，我們研究數種以不同數值通量為基礎的不連續有限元素法求解對流佔優情況下之對流-擴散問題。我們比較了數種不連續有限元素法在Galerkin與Petrov-Galerkin形式下的數值效率，其中所有的不連續Petrov-Galerkin方法皆經由多尺度基函數取代Q1試驗函數而產生，而該多尺度基函數源自於求解各有限單元上具合適邊界條件之局部微分方程式。我們經由兩個具有解析解的數值實例來闡明這些不同方法的效能。我們發現使用Q1試驗函數的不連續有限元素法在擴散係數較小時效率會變差，然而除了Baumann-Oden方法外，其他多尺度不連續Petrov-Galerkin方法都比不連續的有限元素法更能精確捕獲問題解在邊界層的結構性質。

In this thesis, we study various discontinuous finite element methods based on different numerical fluxes for solving convection-diffusion problems with emphasis on the convection-dominated case. We compare numerically the efficiency of various discontinuous finite element methods in the Galerkin and the Petrov-Galerkin formulations. All the discontinuous Petrov-Galerkin methods are formulated by replacing the Q1 trial functions with the multiscale basis functions, which are designed by solving a series of local differential equations on each elements with proper boundary conditions. Numerical simulations of two examples with analytic solutions are presented to illustrate the effectiveness of the various methods. We find that for a small diffusivity, the discontinuous Galerkin methods using Q1 finite elements show a rather poor performance. However, except the Baumann-Oden method, all the other multiscale discontinuous Petrov-Galerkin methods are much better able to capture the nature of boundary layer structure in the solution than the discontinuous Galerkin methods.

[1] D. Arnold, F. Brezzi, B. Cockburn, and D. Marini, Discontinuous Galerkin methods for ellip-
tic problems, in Discontinuous Galerkin Methods: Theory, Computation and Applications,
B. Cockburn, G. E. Karniadakis, and C.-W. Shu, eds., Lecture Notes in Computational
Science and Engineering, Springer-Verlag, New York, 11 (2000), pp. 89-101.
[2] D. N. Arnold, F. Brezzi, B. Cockburn, and D. Marini, Uni¯ed analysis of discontinuous
Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2002), pp.
1749-1779.
[3] I. Babuska and M. Zlamal, Nonconforming elements in the ¯nite element method with
penalty, SIAM Journal on Numerical Analysis, 10 (1973), pp. 863-875.
[4] F. Bassi and S. Rebay, A high-order accurate discontinuous ¯nite element method for the
numerical solution of the compressible Navier-Stokes equations, Journal of Computational
Physics, 131 (1997), pp. 267-279.
[5] F. Bassi and S. Rebay, Numerical evaluation of two discontinuous Galerkin methods for
the compressible Navier-Stokes equations, International Journal for Numerical Methods in
Fluids, 40 (2002), pp. 197-207.
[6] F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, and M. Savini, A high-order accurate discon-
tinuous ¯nite element method for inviscid and viscous turbomachinery °ows, in Proceedings
of 2nd European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics,
Antwerpen, Belgium (1997), Technologisch Instituut, pp. 99-108.
[7] C. E. Baumann and J. T. Oden, A discontinuous hp ¯nite element method for convection-
diRusion problems, Computer Methods in Applied Mechanics and Engineering, 175 (1999),
pp. 311-341.
[8] F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous galerkin approxima-
tions for diRusion problems, in Atti Convegno in onore di F. Brioschi (Milan,1997) Instituto
Lombardo, Accademia di Scienze e Lettere, Milan, Italy, 1999, pp. 197-217.
[9] F. Brezzi, G. Manzini, D. Marini, P. Pietra, and A. Russo, Discontinuous Galerkin ap-
proximations for elliptic problems, Numerical Methods for Partial DiRerential Equations, 16
(2000), pp. 365-378.
[10] B. Cockburn, S. Hou, and C.-W. Shu, The Runge-Kutta local projection discontinuous
Galerkin ¯nite element method for conservation laws, IV. The multidimensional case, Math-
ematics of Computation, 54 (1990), pp. 545-581.
[11] B. Cockburn, G. E. Karniadakis, and C.-W. Shu, The development of discontinuous Galerkin
methods, in Discontinuous Galerkin Methods, Lecture Notes in Computational Science and
Engineering, Springer, Berlin, 11 (2000), pp. 3-50.
[12] B. Cockburn, S. Y. Lin, and C.-W. Shu, The Runge-Kutta local projection discontinuous
Galerkin ¯nite element method for conservation laws, III. One-dimensional systems, Journal
of Computational Physics, 84 (1989), pp. 90-113.
[13] B. Cockburn and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin ¯nite
element method for conservation laws, II. General framework, Mathematics of Computation,
52 (1989), pp. 411-435.
[14] B. Cockburn and C.-W. Shu, The Runge-Kutta local projection P1-discontinuous-Galerkin
¯nite element method for scalar conservation laws, RAIRO. Modelisation Mathematique et
Analyse Numerique, 25 (1991), pp. 337-361.
[15] B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conser-
vation laws, V. Multidimensional systems, Journal of Computational Physics, 141 (1998),
pp. 199-224.
[16] B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent
convection-diRusion systems, SIAM Journal on Numerical Analysis, 35 (1998), pp. 2440-2463.
[17] B. Cockburn and C.-W. Shu, Runge-Kutta discontinuous Galerkin methods for convection
dominated problems, Journal of Scienti¯c Computing, 16 (2001), pp. 173-261.
[18] B. Cockburn, Discontinuous Galerkin methods for convection-dominated problems, in High
order Methods for Computational Physics, Lecture Notes in Computational Science and
Engineering, Springer, Berlin, 9 (1999), pp. 69-224.
[19] J. Douglas and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin
methods, Computer Methods in Applied Sciences, 58 (1976), pp. 207-216.
[20] L. P. Franca, J. V. A. Ramalho, and F. Valentin, Multiscale and residual-free bubble func-
tions for reaction-advection-diRusion problems, International Journal for Multiscale Compu-
tational Engineering, 3 (2005), pp. 297-312.
[21] T. Y. Hou and X.-H. Wu, A multiscale ¯nite element method for elliptic problems in compos-
ite materials and porous media, Journal of Computational Physics, 134 (1997), pp.169-189.
[22] W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation,
Technical report LA-UR-73-479, Los Alamos Scienti¯c Laboratory, Los Alamos, NM, 1973.
[23] B. Riviere, M. F. Wheeler, and V. Girault, Improved energy estimates for interior penalty,
constrained and discontinuous Galerkin methods for elliptic problems I, Computational Geo-
sciences, 3 (1999), pp. 337-360.
[24] J. J. Sudirham, J. J. W. van der Vegt, and R. M. J. van Damme, A study on discontinuous
Galerkin ¯nite element methods for elliptic problems, Memorandum, Department of Applied
Mathematics, University of Twente, Enschede, The Netherland, 1690 (2003).
[25] S. K. Tomar and J. J. W. van der Vegt, A Runge-Kutta discontinuous Galerkin method for
linear free-surface gravity waves using high order velocity recovery, Computer Methods in
Applied Mechanics and Engineering, 196 (2007), pp. 1984-1996.