# Article

Full entry | PDF   (2.5 MB)
Keywords:
degenerate equation; Lipschitz; energy analysis; semi-discrete Galerkin method; semilinear equation; stability; convergence
Summary:
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.
References:
[1] W. F. Ames: Nonlinear Partial Differential in Engineering. Academic Press, New York, 1965. MR 0210342
[1a] J. P. Aubin: Applied Functional Analysis. Wiley-Interscience, New York, 1979. MR 0549483 | Zbl 0424.46001
[2] 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. MR 0421106
[3] G. Baker: Error estimates for finite element methods for second order hyperbolic equations. SIAM J. Numer. Anal., v. 13 (1976), pp. 564-576. DOI 10.1137/0713048 | MR 0423836
[4] G. Baker V. Dougalis: On the $L^\infty$-Convergence of Galerkin approximations for second-order hyperbolic equations. Math. Соmр., v. 34 (1980), pp. 401-424. MR 0559193
[4a] M. L. Bernardi: Second order abstract differential equatiors with singular coefficients. Ann. Math. Рurа Appl., v. 80 (1982), pp. 257-286. DOI 10.1007/BF01761498 | MR 0663974
[5] J. H. Bramble A. H. Schatz V. Thomée L. Wahlbin: Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equatiors. SIAM J. Numer. Anal., v. 14 (1977), pp. 218-241. DOI 10.1137/0714015 | MR 0448926
[6] B. Cahlon: On the initial value problem for a certain partial differential equation. B. I. T., v. 19 (1979), pp. 164-171. MR 0537776 | Zbl 0411.65052
[7] R. W. Carroll R. E. Showalter: Singular and Degenerate Cauchy Problems. Academic Press, New York, 1976. MR 0460842
[7a] R. Courant D. Hilbert: Methods of Mathematical Physics. volume 1, Irterscience Publishers, New York.
[8] J. Douglas, Jr. T. Dupont: Galerkin methods for parabolic equations:. SIAM J. Numer. Anal. v. 7 (1970), pp. 575-626. DOI 10.1137/0707048 | MR 0277126
[9] J. Douglas, Jr. T. Dupont L. Wahlbin: Optimal $L_\infty$ error estimates for Galerkin approximations to two-point boundary value problems. Math. Соmр., v. 29 (1975), pp. 475 - 483. MR 0371077
[10] T. Dupont: $L^2$-Estimates for Galerkin methods for second order hyperbolic equations. SlAM J. Numer. Anal., v. 10 (1973), pp. 880-889. DOI 10.1137/0710073 | MR 0349045
[11] A. Erdélyi W. Magnus F. Oberhettinger F. G. Tricomi: Higher Transcendental Functions. Volume 3. McGraw-Hill, New York, 1953. MR 0058756
[11a] G. Fix N. Nassif: On Finite element approximations to time dependent problems. Numer. Math., v. 19 (1972), pp. 127-135. DOI 10.1007/BF01402523 | MR 0311122
[12] J. Frehse R. Rannacher: Asymptotic $L^\infty$-error estimates for linear finite element approximations of Quasilinear boundary value problems. SIAM J. Numer. Anal., v. 15 (1978), pp. 418-431. DOI 10.1137/0715026 | MR 0502037
[13] A. Friedman Z. Schuss: Degenerate evolution equations in Hilbert space. Trans. Amer. Math. Soc., v. 161 (1971), pp. 401-427. DOI 10.1090/S0002-9947-1971-0283623-9 | MR 0283623
[14] A. Genis: On finite element methods for the Euler-Poisson-Darboux equation. SIAM J. Numer. Anal., to appear. MR 0765508 | Zbl 0576.65112
[15] D. Gilbarg N. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York, 1977. MR 0473443
[16] W. Layton: The finite element method for a degenerate hyperbolic partial differential equation. B.I.T. Zbl 0518.65066
[16a] W. Layton: Some effects of numerical integration in finite element approximations to degenerate evolution equations. Calcolo, to appear. MR 0799613
[17] H. A. Levine: On the nonexistence of global solutions to a nonlinear Euler-Poisson-Darboux equation. J. Math. Anal. and Appl., v. 48 (1974), pp. 646-651. DOI 10.1016/0022-247X(74)90137-1 | MR 0352732 | Zbl 0291.35063
[18] J. A. Nitsche: $L^\infty$-convergence for finite element approximation. Second Conference on Finite Elements. Rennes, France, May 12-14, 1975. MR 0568857
[19] J. A. Nitsche: On $L^\infty$-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. MR 0513215
[19a] M. Povoas: On a second order degenerate hyperbolic equation. Boll. Un. Mat. Ital., v. 5, 16A (1979), pp. 349-355. MR 0541773 | Zbl 0403.35070
[20] M. Reed: Abstract Nonlinear Wave Equations. Sprirger Lecture Notes vol. 507, Springer-Verlag, Berlin, 1970.
[21] R. Scott: Optimal $L^\infty$-estimates for the finite element method on irregular meshes. Math. Соmр., v. 30 (1976), pp. 681-697. MR 0436617
[22] V. Thomée L. Wahlbin: On Galerkin methods in semilinear parabolic problems. SIAM J. Numer. Anal., v. 12 (1975), pp. 378-389. DOI 10.1137/0712030 | MR 0395269
[22a] A. Weinstein: Singular partial differential equations and their applications. pp. 29-49 in Proc. Symp. Fluid Dyn. Appl. Mat. Gordon-Breach, New York, 1962. MR 0153965 | Zbl 0142.07303
[23] M. Zlámal: The mixed boundary value problem for a hyperbolic equation with a small parameter. Czechoslovak Math. J., v. 10 (1960), pp. 83-122.
[24] M. Zlámal: The parabolic equation as a limiting case of a certain elliptic equation. Ann. Mat. Рurа Appl., v. 57 (1962), pp. 143-150. DOI 10.1007/BF02417732 | MR 0138891
[25] M. Zlámal: Finite element multistep discretizations of parabolic boundary value problems. Math. Соmр., v. 29 (1975), pp. 350-359. MR 0371105

Partner of