An energy analysis of degenerate hyperbolic partial differential equations.

William J. Layton

Aplikace matematiky (1984)

  • Volume: 29, Issue: 5, page 350-366
  • ISSN: 0862-7940

Abstract

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An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region Ω (E) ( t u t ) t = ∑ i , j = 1 ( a i j ( x ) u x i ) x j - a 0 ( x ) u + f ( u ) , subject to the initial and boundary conditions, u = 0 on ∂ Ω and u ( x , 0 ) = u 0 . (E) is degenerate at t = 0 and thus, even in the case f ≡ 0 , time derivatives of u will blow up as t → 0 . Also, in the case where f is locally Lipschitz, solutions of (E) can blow up for t > 0 in finite time. Stability and convergence of the scheme in W 2 , 1 is shown in the linear case without assuming u t t (which can blow up as t → 0 is smooth. Convergence of the approximation to u is shown in the case where f is nonlinear and locally Lipschitz. The convergence occurs in regions where u ( x , t ) exists and is smooth. Rates of convergence are given.

How to cite

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Layton, William J.. "An energy analysis of degenerate hyperbolic partial differential equations.." Aplikace matematiky 29.5 (1984): 350-366. <http://eudml.org/doc/15366>.

@article{Layton1984,
abstract = {An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region$\Omega $ (E) $(tu_t)_t=\sum _\{i,j=1\}(a_\{ij\}(x)u_\{x_i\})_\{x_j\} - \{a_0(x)u+f(u)\}$, subject to the initial and boundary conditions, $u=0$ on $\partial \Omega $ and $u(x,0)=u_0$. (E) is degenerate at $t=0$ and thus, even in the case $f\equiv 0$, time derivatives of $u$ will blow up as $t\rightarrow 0$. Also, in the case where $f$ is locally Lipschitz, solutions of (E) can blow up for $t>0$ in finite time. Stability and convergence of the scheme in $W^\{2,1\}$ is shown in the linear case without assuming $u_\{tt\}$ (which can blow up as $t\rightarrow 0$ is smooth. Convergence of the approximation to $u$ is shown in the case where $f$ is nonlinear and locally Lipschitz. The convergence occurs in regions where $u(x,t)$ exists and is smooth. Rates of convergence are given.},
author = {Layton, William J.},
journal = {Aplikace matematiky},
keywords = {degenerate equation; Lipschitz; energy analysis; semi-discrete Galerkin method; semilinear equation; stability; convergence; degenerate equation; Lipschitz; energy analysis; semi-discrete Galerkin method; semilinear equation; Stability; convergence},
language = {eng},
number = {5},
pages = {350-366},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An energy analysis of degenerate hyperbolic partial differential equations.},
url = {http://eudml.org/doc/15366},
volume = {29},
year = {1984},
}

TY - JOUR
AU - Layton, William J.
TI - An energy analysis of degenerate hyperbolic partial differential equations.
JO - Aplikace matematiky
PY - 1984
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 29
IS - 5
SP - 350
EP - 366
AB - An energy analysis is carried out for the usual semidiscrete Galerkin method for the semilinear equation in the region$\Omega $ (E) $(tu_t)_t=\sum _{i,j=1}(a_{ij}(x)u_{x_i})_{x_j} - {a_0(x)u+f(u)}$, subject to the initial and boundary conditions, $u=0$ on $\partial \Omega $ and $u(x,0)=u_0$. (E) is degenerate at $t=0$ and thus, even in the case $f\equiv 0$, time derivatives of $u$ will blow up as $t\rightarrow 0$. Also, in the case where $f$ is locally Lipschitz, solutions of (E) can blow up for $t>0$ in finite time. Stability and convergence of the scheme in $W^{2,1}$ is shown in the linear case without assuming $u_{tt}$ (which can blow up as $t\rightarrow 0$ is smooth. Convergence of the approximation to $u$ is shown in the case where $f$ is nonlinear and locally Lipschitz. The convergence occurs in regions where $u(x,t)$ exists and is smooth. Rates of convergence are given.
LA - eng
KW - degenerate equation; Lipschitz; energy analysis; semi-discrete Galerkin method; semilinear equation; stability; convergence; degenerate equation; Lipschitz; energy analysis; semi-discrete Galerkin method; semilinear equation; Stability; convergence
UR - http://eudml.org/doc/15366
ER -

References

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  1. W. F. Ames, Nonlinear Partial Differential in Engineering, Academic Press, New York, 1965. (1965) MR0210342
  2. J. P. Aubin, Applied Functional Analysis, Wiley-Interscience, New York, 1979. (1979) Zbl0424.46001MR0549483
  3. I. Babuska A. K. Aziz, Survey Lectures on the Mathematical Foundations of the Finite Element Method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. A. K. Aziz (Editor). Academic Press, New York, 1972. (1972) MR0421106
  4. G. Baker, 10.1137/0713048, SIAM J. Numer. Anal., v. 13 (1976), pp. 564-576. (1976) MR0423836DOI10.1137/0713048
  5. G. Baker V. Dougalis, On the L ∞ -Convergence of Galerkin approximations for second-order hyperbolic equations, Math. Соmр., v. 34 (1980), pp. 401-424. (1980) MR0559193
  6. M. L. Bernardi, 10.1007/BF01761498, Ann. Math. Рurа Appl., v. 80 (1982), pp. 257-286. (1982) MR0663974DOI10.1007/BF01761498
  7. J. H. Bramble A. H. Schatz V. Thomée L. Wahlbin, 10.1137/0714015, SIAM J. Numer. Anal., v. 14 (1977), pp. 218-241. (1977) MR0448926DOI10.1137/0714015
  8. B. Cahlon, On the initial value problem for a certain partial differential equation, B. I. T., v. 19 (1979), pp. 164-171. (1979) Zbl0411.65052MR0537776
  9. R. W. Carroll R. E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York, 1976. (1976) MR0460842
  10. R. Courant D. Hilbert, Methods of Mathematical Physics, volume 1, Irterscience Publishers, New York. 
  11. J. Douglas, Jr. T. Dupont, 10.1137/0707048, SIAM J. Numer. Anal. v. 7 (1970), pp. 575-626. (1970) MR0277126DOI10.1137/0707048
  12. J. Douglas, Jr. T. Dupont L. Wahlbin, Optimal L ∞ error estimates for Galerkin approximations to two-point boundary value problems, Math. Соmр., v. 29 (1975), pp. 475 - 483. (1975) MR0371077
  13. T. Dupont, 10.1137/0710073, SlAM J. Numer. Anal., v. 10 (1973), pp. 880-889. (1973) MR0349045DOI10.1137/0710073
  14. A. Erdélyi W. Magnus F. Oberhettinger F. G. Tricomi, Higher Transcendental Functions, Volume 3. McGraw-Hill, New York, 1953. (1953) MR0058756
  15. G. Fix N. Nassif, 10.1007/BF01402523, Numer. Math., v. 19 (1972), pp. 127-135. (1972) MR0311122DOI10.1007/BF01402523
  16. J. Frehse R. Rannacher, 10.1137/0715026, SIAM J. Numer. Anal., v. 15 (1978), pp. 418-431. (1978) MR0502037DOI10.1137/0715026
  17. A. Friedman Z. Schuss, 10.1090/S0002-9947-1971-0283623-9, Trans. Amer. Math. Soc., v. 161 (1971), pp. 401-427. (1971) MR0283623DOI10.1090/S0002-9947-1971-0283623-9
  18. A. Genis, On finite element methods for the Euler-Poisson-Darboux equation, SIAM J. Numer. Anal., to appear. Zbl0576.65112MR0765508
  19. D. Gilbarg N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1977. (1977) MR0473443
  20. W. Layton, The finite element method for a degenerate hyperbolic partial differential equation, B.I.T. Zbl0518.65066
  21. W. Layton, Some effects of numerical integration in finite element approximations to degenerate evolution equations, Calcolo, to appear. MR0799613
  22. H. A. Levine, 10.1016/0022-247X(74)90137-1, J. Math. Anal. and Appl., v. 48 (1974), pp. 646-651. (1974) Zbl0291.35063MR0352732DOI10.1016/0022-247X(74)90137-1
  23. J. A. Nitsche, L ∞ -convergence for finite element approximation, Second Conference on Finite Elements. Rennes, France, May 12-14, 1975. (1975) MR0568857
  24. J. A. Nitsche, On L ∞ -convergence of finite element approximation to the solution of a nonlinear boundary value problem, Topics in Numerical Analysis III, Proc. Roy. Irish Acad. Conf. on Numerical Analysis 1976, J. J. H. Miller (Editor). Academic Press, London, 1977. (1976) MR0513215
  25. M. Povoas, On a second order degenerate hyperbolic equation, Boll. Un. Mat. Ital., v. 5, 16A (1979), pp. 349-355. (1979) Zbl0403.35070MR0541773
  26. M. Reed, Abstract Nonlinear Wave Equations, Sprirger Lecture Notes vol. 507, Springer-Verlag, Berlin, 1970. (1970) 
  27. R. Scott, Optimal L ∞ -estimates for the finite element method on irregular meshes, Math. Соmр., v. 30 (1976), pp. 681-697. (1976) MR0436617
  28. V. Thomée L. Wahlbin, 10.1137/0712030, SIAM J. Numer. Anal., v. 12 (1975), pp. 378-389. (1975) MR0395269DOI10.1137/0712030
  29. A. Weinstein, Singular partial differential equations and their applications, pp. 29-49 in Proc. Symp. Fluid Dyn. Appl. Mat. Gordon-Breach, New York, 1962. (1962) Zbl0142.07303MR0153965
  30. M. Zlámal, The mixed boundary value problem for a hyperbolic equation with a small parameter, Czechoslovak Math. J., v. 10 (1960), pp. 83-122. (1960) 
  31. M. Zlámal, 10.1007/BF02417732, Ann. Mat. Рurа Appl., v. 57 (1962), pp. 143-150. (1962) MR0138891DOI10.1007/BF02417732
  32. M. Zlámal, Finite element multistep discretizations of parabolic boundary value problems, Math. Соmр., v. 29 (1975), pp. 350-359. (1975) MR0371105

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