Abstract
In this paper, we establish the unconditional stability and optimal error estimates of a linearized backward Euler–Galerkin finite element method (FEM) for the time-dependent nonlinear thermistor equations in a two-dimensional nonconvex polygon. Due to the nonlinearity of the equations and the non-smoothness of the solution in a nonconvex polygon, the analysis is not straightforward, while most previous efforts for problems in nonconvex polygons mainly focused on linear models. Our theoretical analysis is based on an error splitting proposed in [30, 31] together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated (time-discrete) elliptic system in a nonconvex polygon. With the proved regularity, we establish the stability in \(l^\infty (L^\infty )\) and the convergence in \(l^\infty (L^2)\) for the fully discrete finite element solution without any restriction on the time-step size. The approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons. Numerical results confirm our theoretical analysis and show clearly that no time-step condition is needed.
Similar content being viewed by others
References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Babuska, I., Kellogg, R.B., Pitkaranta, J.: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33, 447–471 (1979)
Allegretto, W., Xie, H.: Existence of solutions for the time dependent thermistor equation. IMA. J. Appl. Math. 48, 271–281 (1992)
Allegretto, W., Yan, N.: A posteriori error analysis for FEM of thermistor problems. Int. J. Numer. Anal. Model. 3, 413–436 (2006)
Allegretto, W., Lin, Y., Ma, S.: Existence and long time behaviour of solutions to obstacle thermistor equations. Discret. Contin. Dyn. Syst. Ser. A 8, 757–780 (2002)
Bacuta, C., Bramble, J.H., Xu, J.: Regularity estimates for elliptic boundary value problems in Besov spaces. Math. Comp. 72, 1577–1595 (2003)
Bernardi, C., Dauge, M., Maday, Y.: Polynomials in the Sobolev World. IRMAR 07-14, Rennes, March 2007 (Preprint)
Byun, S.S., Wang, L.: Elliptic equations with measurable coefficients in Reifenberg domains. Adv. Math. 225, 2648–2673 (2010)
Cai, Z., Kim, S.: A finite element method using singular functions for the Poisson equation: corner singularities. SIAM J. Numer. Anal. 39, 286–299 (2001)
Cai, Z., Kim, S., Shin, B.C.: Solution methods for the Poisson equation with corner singularities: numerical results. SIAM J. Sci. Comput. 23, 672–682 (2001)
Chatzipantelidis, P., Lazarov, R.D., Thomée, V., Wahlbin, L.B.: Parabolic finite element equations in nonconvex polygonal domains. BIT Numer. Math. 46, S113–S143 (2006)
Chatzipantelidis, P., Lazarov, R.D., Thomée, V.: Parabolic finite volume element equations in nonconvex polygonal domains. Numer. Methods Partial Differ. Equ. 25, 507–525 (2009)
Chrysafinos, K., Hou, L.S.: Error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions. SIAM J. Numer. Anal. 40, 282–306 (2002)
Cimatti, G.: Existence of weak solutions for the nonstationary problem of the joule heating of a conductor. Ann. Mat. Pura Appl. 162, 33–42 (1992)
Chen, Y.-Z., Wu, L.-C.: Second Order Elliptic Equations and Elliptic Systems, Translations of Mathematical Monographs, vol. 174. AMS, Providence (1998)
Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Springer, New York (1988)
Elliott, C.M., Larsson, S.: A finite element model for the time-dependent joule heating problem. Math. Comput. 64, 1433–1453 (1995)
Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)
Geuzaine, C., Remacle, J.F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79, 1309–1331 (2009)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Pending Action. SIAM, Philadelphia (2011)
Grisvard, P.: Singularities in Boundary Value Problems, Rech. Math. Appl., vol. 22, Masson, Paris (1992)
Guo, B.Q., Schwab, C.: Analytic regularity of Stokes flow on polygonal domains in countably weighted Sobolev spaces. J. Comput. Appl. Math. 190, 487–519 (2006)
Guo, B.Q., Babuska, I.: Regularity of the solutions for elliptic problems on nonsmooth domains in \(^3\). I. Countably normed spaces on polyhedral domains. Proc. R. Soc. Edinb. Sect. A 127, 77–126 (1997)
Jerison, D., Kenig, C.E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161 (1995)
Kellogg, B.: Interpolation between subspaces of a Hilbert space. Institute for Fluid Dynamics and Applied Mathematics, College Park, Technical note (1971)
Kweon, J.R.: The evolution compressible Navier–Stokes system on polygonal domains. J. Differ. Equ. 232, 487–520 (2007)
Kweon, J.R.: Regularity of Solutions for the Navier–Stokes system of incompressible flows on a polygon. J. Differ. Equ. 235, 166–198 (2007)
Ladyzenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasilinear Equations of Parabolic Typez, Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968)
Li, B., Gao, H., Sun, W.: Unconditionally optimal error estimates of a Crank–Nicolson Galerkin method for the nonlinear thermistor equations. SIAM J. Numer. Anal. 52, 933–954 (2014)
Li, B., Sun, W.: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. Model. 10, 622–633 (2013)
Li, B., Sun, W.: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media. SIAM J. Numer. Anal. 51, 1959–1977 (2013)
Li, B., Yang, C.: Uniform BMO estimate of parabolic equations and global well-posedness of the thermistor problem. Forum Math. Sigma 3, e26 (2015). doi:10.1017/fms.2015.29
Li, H.: Finite element analysis for the axisymmetric Laplace operator on polygonal domains. J. Comput. Appl. Math. 235, 5155–5176 (2011)
Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comp. 38, 437–445 (1982)
Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. AMS, Providence (1997)
Simader, C.G.: On Dirichlet Boundary Value Problem: Lp Theory Based on a Generalization of Garding’s Inequality, Lecture Notes in Math, vol. 268. Springer, Berlin (1972)
Schatz, A.H., Wahlbin, L.B.: Maximum norm estimates in the finite element method on plane polygonal domainsz, Part 1. Math. Comput. 32, 73–109 (1978)
Sun, W., Sun, Z.: Finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials. Numer. Math. 120, 153–187 (2012)
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berkub (1997)
Wahlbin, L.B.: On the sharpness of certain local estimates for \(H^1\) projections into finite element spaces: influence of a re-entrant corner. Math. Comput. 42, 1–8 (1984)
Wood, I.: Maximal \(L^p\)- regularity for the Laplacian on Lipschitz domains. Math. Z. 255, 855–875 (2007)
Yue, X.Y.: Numerical analysis of nonstationary thermistor problem. J. Comput. Math. 12, 213–223 (1994)
Yuan, G.: Local existence of bounded solutions to the degenerate Stefan problem with Joule’s heating. J. Partial Differ. Equ. 9, 42–54 (1996)
Yuan, G.: Regularity of solutions of the thermistor problem. Appl. Anal. 53, 149–155 (1994)
Yuan, G., Liu, Z.: Existence and uniqueness of the \(C^\alpha \) solution for the thermistor problem with mixed boundary value. SIAM J. Math. Anal. 25, 1157–1166 (1994)
Zhao, W.: Convergence analysis of finite element method for the nonstationary thermistor problem. Shandong Daxue Xuebao 29, 361–367 (1994)
Zhou, S., Westbrook, D.R.: Numerical solutions of the thermistor equations. J. Comput. Appl. Math. 79, 101–118 (1997)
Acknowledgments
The work of H. Gao was supported in part by NSFC 11501227. The work of B. Li was partially supported by the Start-up Fund (A/C Code: 1-ZE6L) of The Hong Kong Polytechnic University, and was partially carried during a research stay at Universität Tübingen, supported by the Alexander von Humboldt Foundation and NSFC 11301262. The research of W. Sun was supported in part by a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11302915).
Author information
Authors and Affiliations
Corresponding author
Appendix: Proof of Lemma 1
Appendix: Proof of Lemma 1
The following lemmas are consequences of [24] (which can be extended to Hölder continuous coefficients via a basic perturbation argument) and [41].
Lemma A.1
Let \(\Omega \) be a Lipschitz domain in \(\mathbb {R}^2\). Suppose that \(\sigma (u)\) is Hölder continuous and satisfies (1.5). Then there exists a positive constant \(p_1>4\) (depending on the domain \(\Omega \)) such that the solution v of the equation
satisfies that
Lemma A.2
Let \(\Omega \) be a Lipschitz domain in \(\mathbb {R}^2\). Then the solution of the inhomogeneous heat equation
satisfies that
From [7] we know that the regularity of g given in Lemma 1 implies that g can be extended to the interior of the domain \(\Omega \) with \(g\in L^\infty ((0,T);H^{1+\beta })\) and \(g_t\in L^\infty ((0,T);W^{1,4})\).
Based on Yuan and Liu’s results [44, 45], the solution of (1.2)–(1.4) satisfies that
By Lemma A.1, the Eq. (1.3) with the Hölder continuity of u implies that
Then (1.2) implies that
Let \(w=u_t\). Differentiating (1.2)–(1.3) with respect to t, we obtain
with the initial condition \(w(x,0)=w_0(x)\), where \(w_0=\varDelta u_0+\sigma (u_0)|\nabla \phi _0|^2\) and \(\phi _0\) is the solution of the elliptic PDE
It follows that \(w_0\in H^{1}_0\). Since \(w\in L^2((0,T); L^2)\) and \(\nabla \phi \in L^\infty ((0,T); L^4)\), applying Lemma A.1–(A.5) gives
and by the Sobolev embedding theorem, we obtain,
Again applying Lemma A.2 to (A.4) shows
and with the Sobolev embedding theorem, we have
From (A.5) and (A.4), we see that
and
By the Sobolev embedding theorem, we have further
where \(p_3\) is determined by \(1/p_3+1/p_1=1/4\). Then
Moreover, by applying Lemma A.1 to the Eq. (A.5), we see that
and by Lemma A.2, (A.4) implies that
Using the Sobolev embedding theorem again, we arrive at
Differentiating (1.2) with respect to t, we get
and the above equation times \(\varDelta w\) gives
which further shows that
It follows that
From (A.5) we see that
Furthermore, we rewrite (1.3) by
With (), we obtain
Since \(H^{1+s_1}\hookrightarrow \hookrightarrow H^{1+s}\hookrightarrow W^{1,4}\), we have the following estimate (see Lemma 1.1, pp. 106 of [35])
which implies that \(\phi \in C([0,T];H^{1+s})\).
The proof of Lemma 1 is completed. \(\square \)
Rights and permissions
About this article
Cite this article
Gao, H., Li, B. & Sun, W. Stability and convergence of fully discrete Galerkin FEMs for the nonlinear thermistor equations in a nonconvex polygon. Numer. Math. 136, 383–409 (2017). https://doi.org/10.1007/s00211-016-0843-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-016-0843-9
Keywords
- Finite element method
- Nonconvex polygon
- Unconditional stability
- Optimal error estimate
- Thermistor problem