Finite-difference Approximation of Mathematical Physics Problems on Irregular Grids

Computational Methods in Applied Mathematics

Volumn 5, Issue 3, 2005, Pages 294-330

  Vabishchevich, P N ^a  

^a INSTITUTE OF MATHEMATICAL MODELING   (Russian Federation)

Author keywords

convection diffusion problems; Delaunay triangulation; finite difference method; unstructured grids; vector problems; Voronoi diagrams

Indexed keywords

EID: 35448959458     PISSN: 16094840     EISSN: 16099389     Source Type: Journal    
DOI: 10.2478/cmam-2005-0015     Document Type: Article

References (142)

1. N. V. Ardelyan, The convergence of difference schemes for two-dimensional equations of acoustics and maxwells equations, J. of Comput. Math. and Math. Phys., 23 (1983), no. 5, pp. 93-99.
2. N. V. Ardelyan and S. V. Chernigovskii, Convergence of difference schemes for two-dimensional gas-dynamic equations in acoustics approximations with gravitation taken into account, Differential Equations, 20 (1984), no. 7, pp. 807-813.
3. N. V. Ardelyan and K. V. Kosmachevskii, Implicit arbitrary-lagrange method for predicting 2d magneto-gasdinamic flows, in: Mathematical Modeling, Moscow University Publishers, Moscow, 1993, pp. 25-44.
4. F. Assous, P. Degond, E. Heintze, P. A. Raviart, and J. Segre, On a finite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys., 109 (1993), no. 2, pp. 222-237.
5. R. T. Atkinson and H. Weimin, Theoretical Numerical Analysis. A Functional Analysis Framework, vol. 39 of Texts in Applied Mathematics, Springer, 2001.
6. F. Aurenhammer, Voronoi diagrams-a survey of a fundamental geometric data structure, ACM Comput. Surveys, 23 (1991), pp. 345-405.
7. F. Aurenhammer and R. Klein, Voronoi diagrams, in: Handbook of Computational Geometry (J. Sack and G. Urrutia, eds.), Elsevier Science Publishing, 2000, pp. 201-290.
8. A. Bossavit, Computational electromagnetism, Academic Press Inc., San Diego, CA, 1998.
9. A. Bossavit, 'Generalized finite differences' in computational electromagnetics, in: Geometric Methods in Computational Electromagnetics, PIER 32 (F. L. Teixeira, ed.), EMW Publishing, Cambridge, Mass., 2001, pp. 45-64.
10. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York - Berlin - Heidelberg, 1991.
11. P. Chatzipantelidis, R. D. Lazarov, and V. Thomee, Error estimates for the finite volume element method for parabolic equations in convex polygonal domains, Numerical Methods for Partial Differential Equations, 20 (2004), pp. 650-674.
12. S.-H. Chou and Q. Li, Error estimates in l2, h1 and lx in covolume methods for elliptic and parabolic problems: A unified approach, Math. Comp., 69 (2000), pp. 103-120.
13. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, 1978.
14. R. Dautray and J. Lions, Mathematical analysis and numerical methods for science and technology, Springer, Berlin, 1988-1993.
15. R. E. Ewing, T. Lin, and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal., 39 (2002), no. 6, pp. 1865-1888.
16. R. Eymard, T. Gallouet, and R. Herbin, Finite volume methods, in: Handbook of Numerical Analysis, vol. 7, North Holland, Amsterdam, 2000, pp. 713-1020.
17. A. P. Favorskii, A. A. Samarskii, M. Y. Shashkov, and V. F. Tishkin, Operational finite-difference schemes, Differential Equations, 17 (1981), pp. 854-862.
18. S. Fortune, Voronoi diagrams and Delaunay triangulations, in: Computing in Euclidean Geometry, vol. 1 of Lecture Notes Series of Computing, 1992, pp. 193-233.
19. V. Ganzha, R. Liska, M. Shashkov, and C. Zenger, Support operator method for Laplace equation on unstructured triangular grid, Selcuk J. Appl. Math., 3 (2002), no. 1, pp. 21-48.
20. P.-L. George and H. Borouchaki, Delaunay Triangulation and Meshing: Application to Finite Elements, Hermes, Paris, 1998.
21. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, New York, 1986.
22. B. Heinrich, Finite difference methods on irregular networks, vol. 82 of Internationale Schriftenreihe zur Numerischen Mathematik [International Series of Numerical Mathematics], Birkhauser Verlag, Basel, 1987.
23. F. Hermeline, Two coupled particle-finite volume methods using delaunay-voronoi meshes for the approximation of vlasov-poisson and vlasov-maxwell equations, J. Comput. Phys., 106 (1993), no. 1, pp. 1-18.
24. R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, (2002), pp. 237-339.
25. J. Hyman and M. Shashkov, Mimetic finite difference methods for Maxwell's equations and the equations of magnetic diffusion, in: Geometric Methods in Computational Electromagnetics, PIER 32 (F. L. Teixeira, ed.), EMW Publishing, Cambridge, Mass., 2001, pp. 89-121.
26. J. M. Hyman and M. J. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Comput. Math. Appl., 33 (1997), no. 4, pp. 81-104.
27. J. M. Hyman and M. J. Shashkov, The orthogonal decomposition theorems for mimetic finite difference methods, SIAM J. Numer. Anal., 36 (1999), no. 3, pp. 788-818.
28. B. N. Jiang, J. Wu, and L. A. Povinelli, The origin of spurious solutions in computational electromagnetics, J. Comput. Phys., 125 (1996), no. 1, pp. 104-123.
29. L. D. Landau and E. M. Lifschitz, Fluid Mechanics, Pergamon Press, Oxford, 1982.
30. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984.
31. M. J. Laslo, Computational Geometry and Computer Graphics in C++, Prentice Hall, 1996.
32. R. D. Lazarov and P. S. Vassilevski, Numerical methods for convection-diffusion problems on general grids, in: Approximation Theory: A volume dedicated to Blogovest Sendov (B. Bojanov, ed.), DARBA, Sofia, 2002, pp. 258-283.
33. V. I. Lebedev, Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. i, J. of Comput. Math. and Math. Phys., 4 (1964), no. 3, pp. 69-92, in Russian.
34. V. I. Lebedev, Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. ii, J. of Comput. Math. and Math. Phys., 4 (1964), no. 4, pp. 36-50, in Russian.
35. R. Li, Z. Chen, and W. Wu, Generalized Difference Methods for Differencial Equations. Numerical Analysis of Finite Volume Methods, vol. 226 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 2000.
36. N. K. Madsen, Divergence preserving discrete surface integral methods for Maxwell's curl equations using non-orthogonal unstructured grids, J. Comput. Phys., 119 (1995), pp. 34-45.
37. N. K. Madsen and R. W. Ziolkowski, A three-dimensional modified finite volume technique for Maxwell's equations, Electromagnetics, (1990), no. 1/2, pp. 147-161.
38. M. Marrone, Computational aspects of the cell method in electrodynamics, in: Geometric Methods in Computational Electromagnetics, PIER 32 (F. L. Teixeira, ed.), EMW Publishing, Cambridge, Mass., 2001, p. 317-356.
39. I. D. Mishev, Finite volume methods on voronoi meshes, Numerical Methods for Partial Differential Equations, 14 (1998), no. 2, pp. 193-212.
40. P. Monk and E. Süli, A convergence analysis of Yee's scheme on non-uniform grids, SIAM Journal of Numerical Analysis, 31 (1994), no. 2.
41. K. W. Morton, Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, 1996.
42. J.-C. Nedelec, Mixed finite elements in r3, Numer. Math., 35 (1980), no. 3, pp. 315-341.
43. R. A. Nicolaides and D.-Q. Wang, Convergence analysis of a covolume scheme for maxwell's equations in three dimensions, Math. Comp., 67 (1998), no. 223, pp. 947-963.
44. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, Interscience Publisher, John Wiley & Sons, New York, 1967.
45. N. Robidoux and S. Steinberg, A discrete vector calculus in tensor grids, 2002.
46. M. N. O. Sadiku, Numerical Techniques in Electromagnetics, CRC Press, Boca Raton, 1992.
47. A. Samarskii and I. Frjazinov, Difference approximation methods for problems of mathematical physics, Russ. Math. Surv., 31 (1976), no. 6, pp. 179-213.
48. A. A. Samarskii, Parabolic equations and difference methods for their solution, in: Proc. of All-Union Conference on Differential Equations, 1958, Publishers of Armenian Ac. Sci., 1960, pp. 148-160.
49. A. A. Samarskii, The theory of difference schemes, vol. 240 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 2001.
50. A. A. Samarskii and A. V. Gulin, Stability of difference schemes, Moscow, 1973, in Russian.
51. A. A. Samarskii, A. V. Koldoba, Y. A. Povechenko, V. F. Tishkin, and A. P. Favorskii, Difference Schemes on Unstructured Grids, Kriterii, Minsk, 1996, in Russian.
52. A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors, Kluwer Academic Publishers, Dordrecht Hardbound, 2002.
53. A. A. Samarskii and A. P. Mikhailov, Principles of Mathematical Modeling. Ideas, Methods, Examples, Taylor and Francis, London and New York, 2002.
54. A. A. Samarskii and Y. P. Popov, Difference Methods for the Solution of Problems of Gas Dynamics, Nauka, Moscow, 1992, in Russian.
55. A. A. Samarskii and P. N. Vabishchevich, Additive Schemes for Problems of Mathematical Physics, Nauka, Moscow, 1999, in Russian.
56. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems, Editorial URSS, Moscow, 2003, in Russian.
57. A. A. Samarskii, V. F. Tishkin, A. P. Favorskii, and M. Y. Shashkov, Use of the support operator method for constructing difference analogues of operations of tensor analysis, Differential Equations, 18 (1982), no. 7, pp. 1251-1256.
58. M. J. Shashkov, Conservative finite-difference methods on general grids, CRC Press, Boca Raton, FL, 1996.
59. K. L. Shlager and J. B. Schneider, A survey of the finite-difference time-domain literature, in: Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (A. Taflove, ed.), Artech House, Boston, MA, 1998, pp. 1-62.
60. A. V. Solov'ev, E. V. Solov'eva, V. F. Tishkin, A. P. Favorskij, and M. Shashkov, Approximation of finite-difference operators on a mesh of dirichlet cells, Differential Equations, 22 (1986), no. 7, pp. 863-872.
61. G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N. J., 1973.
62. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Chapman and Hall, 1989.
63. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2 ed., Artech House, Boston, MA, 2000.
64. V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, New York, 1997.
65. J. F. Thompson, B. K. Soni, and N. P. Weatherill, Grid Generation, CRC Press, Boca Raton, 1999.
66. G. Toth, The div b=0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161 (2000), no. 2, pp. 605-652.
67. P. N. Vabishchevich, Difference schemes for nonstationary vector methods, Differential Equations, 40 (2004), no. 7, pp. 1000-1008.
68. P. N. Vabishchevich and A. A. Samarskii, Difference schemes for convection-diffusion problems on irregular grids, J. of Comput. Math. and Math. Phys., 40 (2000), no. 5, pp. 726-739, in Russian.
69. H. K. Versteeg and W. Malalaseker, An introduction to Computational Fluid Dynamics: The Finite Volume Method, Longman Scientific & Technical, 1995.
70. K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, 14, (1966), no. 3, pp. 302-307.
71. O. C. Zienkiewicz, The Finite Element Method, McGraw Hill, London, 1977.
72. N. V. Ardelyan, The convergence of difference schemes for two-dimensional equations of acoustics and maxwells equations, J. of Comput. Math. and Math. Phys., 23 (1983), no. 5, pp. 93-99.
73. N. V. Ardelyan and S. V. Chernigovskii, Convergence of difference schemes for two-dimensional gas-dynamic equations in acoustics approximations with gravitation taken into account, Differential Equations, 20 (1984), no. 7, pp. 807-813.
74. N. V. Ardelyan and K. V. Kosmachevskii, Implicit arbitrary-lagrange method for predicting 2d magneto-gasdinamic flows, in: Mathematical Modeling, Moscow University Publishers, Moscow, 1993, pp. 25-44.
75. F. Assous, P. Degond, E. Heintze, P. A. Raviart, and J. Segre, On a finite-element method for solving the three-dimensional Maxwell equations, J. Comput. Phys., 109 (1993), no. 2, pp. 222-237.
76. R. T. Atkinson and H. Weimin, Theoretical Numerical Analysis. A Functional Analysis Framework, vol. 39 of Texts in Applied Mathematics, Springer, 2001.
77. F. Aurenhammer, Voronoi diagrams - a survey of a fundamental geometric data structure, ACM Comput. Surveys, 23 (1991), pp. 345-405.
78. F. Aurenhammer and R. Klein, Voronoi diagrams, in: Handbook of Computational Geometry (J. Sack and G. Urrutia, eds.), Elsevier Science Publishing, 2000, pp. 201-290.
79. A. Bossavit, Computational electromagnetism, Academic Press Inc., San Diego, CA, 1998.
80. A. Bossavit, 'Generalized finite differences' in computational electromagnetics, in: Geometric Methods in Computational Electromagnetics, PIER 32 (F. L. Teixeira, ed.), EMW Publishing, Cambridge, Mass., 2001, pp. 45-64.
81. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York - Berlin - Heidelberg, 1991.
82. P. Chatzipantelidis, R. D. Lazarov, and V. Thomee, Error estimates for the finite volume element method for parabolic equations in convex polygonal domains, Numerical Methods for Partial Differential Equations, 20 (2004), pp. 650-674.
83. S.-H. Chou and Q. Li, Error estimates in l2, h1 and lx in covolume methods for elliptic and parabolic problems: A unified approach, Math. Comp., 69 (2000), pp. 103-120.
84. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, 1978.
85. R. Dautray and J. Lions, Mathematical analysis and numerical methods for science and technology, Springer, Berlin, 1988-1993.
86. R. E. Ewing, T. Lin, and Y. Lin, On the accuracy of the finite volume element method based on piecewise linear polynomials, SIAM J. Numer. Anal., 39 (2002), no. 6, pp. 1865-1888.
87. R. Eymard, T. Gallouet, and R. Herbin, Finite volume methods, in: Handbook of Numerical Analysis, vol. 7, North Holland, Amsterdam, 2000, pp. 713-1020.
88. A. P. Favorskii, A. A. Samarskii, M. Y. Shashkov, and V. F. Tishkin, Operational finite-difference schemes, Differential Equations, 17 (1981), pp. 854-862.
89. S. Fortune, Voronoi diagrams and Delaunay triangulations, in: Computing in Euclidean Geometry, vol. 1 of Lecture Notes Series of Computing, 1992, pp. 193-233.
90. V. Ganzha, R. Liska, M. Shashkov, and C. Zenger, Support operator method for Laplace equation on unstructured triangular grid, Selcuk J. Appl. Math., 3 (2002), no. 1, pp. 21-48.
91. P.-L. George and H. Borouchaki, Delaunay Triangulation and Meshing: Application to Finite Elements, Hermes, Paris, 1998.
92. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, New York, 1986.
93. B. Heinrich, Finite difference methods on irregular networks, vol. 82 of Internationale Schriftenreihe zur Numerischen Mathematik [International Series of Numerical Mathematics], Birkhauser Verlag, Basel, 1987.
94. F. Hermeline, Two coupled particle-finite volume methods using delaunay-voronoi meshes for the approximation of vlasov-poisson and vlasov-maxwell equations, J. Comput. Phys., 106 (1993), no. 1, pp. 1-18.
95. R. Hiptmair, Finite elements in computational electromagnetism, Acta Numerica, (2002), pp. 237-339.
96. J. Hyman and M. Shashkov, Mimetic finite difference methods for Maxwell's equations and the equations of magnetic diffusion, in: Geometric Methods in Computational Electromagnetics, PIER 32 (F. L. Teixeira, ed.), EMW Publishing, Cambridge, Mass., 2001, pp. 89-121.
97. J. M. Hyman and M. J. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Comput. Math. Appl., 33 (1997), no. 4, pp. 81-104.
98. J. M. Hyman and M. J. Shashkov, The orthogonal decomposition theorems for mimetic finite difference methods, SIAM J. Numer. Anal., 36 (1999), no. 3, pp. 788-818.
99. B. N. Jiang, J. Wu, and L. A. Povinelli, The origin of spurious solutions in computational electromagnetics, J. Comput. Phys., 125 (1996), no. 1, pp. 104-123.
100. L. D. Landau and E. M. Lifschitz, Fluid Mechanics, Pergamon Press, Oxford, 1982.
101. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984.
102. M. J. Laslo, Computational Geometry and Computer Graphics in C++, Prentice Hall, 1996.
103. R. D. Lazarov and P. S. Vassilevski, Numerical methods for convection-diffusion problems on general grids, in: Approximation Theory: A volume dedicated to Blogovest Sendov (B. Bojanov, ed.), DARBA, Sofia, 2002, pp. 258-283.
104. V. I. Lebedev, Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. i, J. of Comput. Math. and Math. Phys., 4 (1964), no. 3, pp. 69-92, in Russian.
105. V. I. Lebedev, Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. ii, J. of Comput. Math. and Math. Phys., 4 (1964), no. 4, pp. 36-50, in Russian.
106. R. Li, Z. Chen, and W. Wu, Generalized Difference Methods for Differencial Equations. Numerical Analysis of Finite Volume Methods, vol. 226 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 2000.
107. N. K. Madsen, Divergence preserving discrete surface integral methods for Maxwell's curl equations using non-orthogonal unstructured grids, J. Comput. Phys., 119 (1995), pp. 34-45.
108. N. K. Madsen and R. W. Ziolkowski, A three-dimensional modified finite volume technique for Maxwell's equations, Electromagnetics, (1990), no. 1/2, pp. 147-161.
109. M. Marrone, Computational aspects of the cell method in electrodynamics, in: Geometric Methods in Computational Electromagnetics, PIER 32 (F. L. Teixeira, ed.), EMW Publishing, Cambridge, Mass., 2001, p. 317-356.
110. I. D. Mishev, Finite volume methods on voronoi meshes, Numerical Methods for Partial Differential Equations, 14 (1998), no. 2, pp. 193-212.
111. P. Monk and E. Süli, A convergence analysis of Yee's scheme on non-uniform grids, SIAM Journal of Numerical Analysis, 31 (1994), no. 2.
112. K. W. Morton, Numerical Solution of Convection-Diffusion Problems, Chapman & Hall, 1996.
113. J.-C. Nedelec, Mixed finite elements in r3, Numer. Math., 35 (1980), no. 3, pp. 315-341.
114. R. A. Nicolaides and D.-Q. Wang, Convergence analysis of a covolume scheme for maxwell's equations in three dimensions, Math. Comp., 67 (1998), no. 223, pp. 947-963.
115. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, Interscience Publisher, John Wiley & Sons, New York, 1967.
116. N. Robidoux and S. Steinberg, A discrete vector calculus in tensor grids, 2002.
117. M. N. O. Sadiku, Numerical Techniques in Electromagnetics, CRC Press, Boca Raton, 1992.
118. A. Samarskii and I. Frjazinov, Difference approximation methods for problems of mathematical physics, Russ. Math. Surv., 31 (1976), no. 6, pp. 179-213.
119. A. A. Samarskii, Parabolic equations and difference methods for their solution, in: Proc. of All-Union Conference on Differential Equations, 1958, Publishers of Armenian Ac. Sci., 1960, pp. 148-160.
120. A. A. Samarskii, The theory of difference schemes, vol. 240 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York, 2001.
121. A. A. Samarskii and A. V. Gulin, Stability of difference schemes, Moscow, 1973, in Russian.
122. A. A. Samarskii, A. V. Koldoba, Y. A. Povechenko, V. F. Tishkin, and A. P. Favorskii, Difference Schemes on Unstructured Grids, Kriterii, Minsk, 1996, in Russian.
123. A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors, Kluwer Academic Publishers, Dordrecht Hardbound, 2002.
124. A. A. Samarskii and A. P. Mikhailov, Principles of Mathematical Modeling. Ideas, Methods, Examples, Taylor and Francis, London and New York, 2002.
125. A. A. Samarskii and Y. P. Popov, Difference Methods for the Solution of Problems of Gas Dynamics, Nauka, Moscow, 1992, in Russian.
126. A. A. Samarskii and P. N. Vabishchevich, Additive Schemes for Problems of Mathematical Physics, Nauka, Moscow, 1999, in Russian.
127. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Convection-Diffusion Problems, Editorial URSS, Moscow, 2003, in Russian.
128. A. A. Samarskii, V. F. Tishkin, A. P. Favorskii, and M. Y. Shashkov, Use of the support operator method for constructing difference analogues of operations of tensor analysis, Differential Equations, 18 (1982), no. 7, pp. 1251-1256.
129. M. J. Shashkov, Conservative finite-difference methods on general grids, CRC Press, Boca Raton, FL, 1996.
130. K. L. Shlager and J. B. Schneider, A survey of the finite-difference time-domain literature, in: Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (A. Taflove, ed.), Artech House, Boston, MA, 1998, pp. 1-62.
131. A. V. Solov'ev, E. V. Solov'eva, V. F. Tishkin, A. P. Favorskij, and M. Shashkov, Approximation of finite-difference operators on a mesh of dirichlet cells, Differential Equations, 22 (1986), no. 7, pp. 863-872.
132. G. Strang and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N. J., 1973.
133. J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Chapman and Hall, 1989.
134. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2 ed., Artech House, Boston, MA, 2000.
135. V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, New York, 1997.
136. J. F. Thompson, B. K. Soni, and N. P. Weatherill, Grid Generation, CRC Press, Boca Raton, 1999.
137. G. Toth, The div b=0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161 (2000), no. 2, pp. 605-652.
138. P. N. Vabishchevich, Difference schemes for nonstationary vector methods, Differential Equations, 40 (2004), no. 7, pp. 1000-1008.
139. P. N. Vabishchevich and A. A. Samarskii, Difference schemes for convection-diffusion problems on irregular grids, J. of Comput. Math. and Math. Phys., 40 (2000), no. 5, pp. 726-739, in Russian.
140. H. K. Versteeg and W. Malalaseker, An introduction to Computational Fluid Dynamics: The Finite Volume Method, Longman Scientific & Technical, 1995.
141. K. S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media, 14, (1966), no. 3, pp. 302-307.
142. O. C. Zienkiewicz, The Finite Element Method, McGraw Hill, London, 1977.