Abstract
We investigate some theoretical and numerical aspects of a nonlinear wave equation with variable coefficient and Dirichlet and Acoustic boundary conditions. The existence and uniqueness of the solution are obtained applying the Faedo-Galerkin method with some compactness results and energy method. In addition, we prove the uniform stability of the energy. For numerical simulation, firstly we use the Crank–Nicolson Galerkin method, in which it consists of applying the finite element method in the spatial variable and the Crank–Nicolson method over time. Subsequently, in the resulting nonlinear algebraic system, for each discrete time, we apply the Newton’s method without losing the convergence order. Moreover, are presented figures of the numerical solutions for the two-dimensional case, tables with error and convergence order and the numerical energy decay. These results justify the consistency between the theoretical and numerical results.
Similar content being viewed by others
References
Alcântara AA, Clark HR, Rincon MA (2018) Theoretical analysis and numerical simulation for a hyperbolic equation with dirichlet and acoustic boundary conditions. Comput Appl Math 37(4):4772–4792
Baker GA (1976) Error estimates for finite element methods for second order hyperbolic equations. SIAM J Numer Anal 13:564–576
Basson M, van Rensburg NFJ (2013) Galerkin finite element approximation of general linear second order hyperbolic equations. Numer Funct Anal Optim 34(9):976–1000
Beale JT (1976) Spectral properties of an acoustic boundary condition. Indiana Univ Math J 25(9):895–917
Beale JT (1977) Acoustic scattering from locally reacting surfaces. Indiana Univ Math J 26(2):199–222
Beale JT, Rosencrans SI (1974) Acoustic boundary conditions. Bull Am Math Soc 80(6):1276–1278
Carmo BA, Clark HR, Guardia RR, Rincon MA (2018) Mathematical analysis and numerical simulation of a nonlinear thermoelastic system. Numer Funct Anal Optim pp 1–29
Cavalcanti MM, Domingos Cavalcanti VN, Frota CL, Vicente A (2020) Stability for semilinear wave equation in an inhomogeneous medium with frictional localized damping and acoustic boundary conditions. SIAM J Control Optim 58(4):2411–2445
Cavalcanti M SJ Domingos Cavalcanti V (2006) Global Solvability and Asymptotic Stability for the Wave Equation with Nonlinear Boundary Damping and Source Term, Birkhäuser Basel, pp 161–184
Chen G (1979) Energy decay estimates and exact boundary-value controllability for the wave equation in a bounded domain’’. Journal de Mathématiques Pures et Appliquées 58:249–273
Chen Y, Huang Y (1998) The full-discrete mixed finite element methods for nonlinear hyperbolic equations. Commun Nonlinear Sci Numer Simul 3(3):152–155
Chrysafinos K, Hou LS (2017) Semi-discrete error estimates of the evolutionary stokes equations with inhomogeneous dirichlet boundary data. Comput Math Appl 73(8):1684–1696
Ciarlet PG (2002) The finite element method for elliptic problems. SIAM
Clark H, Jutuca LSG, Miranda MM (1998) On a mixed problem for a linear coupled system with variable coefficients. Eletron J Differ Equ 1(4):1–20
Cousin AT, Frota CL, Larkin NA (2004) On a system of klein-gordon type equations with acoustic boundary conditions. J Math Anal Appl 293(1):293–309
Dupont T (1973) \({L}^2\)-estimates for galerkin methods for second order hyperbolic equations. SIAM J Numer Anal 10(5):880–889
Frigeri S (2010) Attractors for semilinear damped wave equations with an acoustic boundary condition. J Evol Equ 10(1):29–58
Frota C, Medeiros L, Vicente A et al (2011) Wave equation in domains with non-locally reacting boundary. Differ Integr Equ 24(11/12):1001–1020
Frota CL, Goldstein JA (2000) Some nonlinear wave equations with acoustic boundary conditions. J Differ Equ 164(1):92–109
Frota CL, Larkin NA (2005) Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. In: Contributions to nonlinear analysis, Springer, pp 297–312
Graber PJ (2012) Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping. J Evol Equ 12(1):141–164
Gunzburger M, Hou S (1992) Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses. SIAM J Numer Anal 29(2):390–424
Ha TG (2016) General decay estimates for the wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary. Appl Math Lett 60:43–49
Hipp D, Kovács B (2020) Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates. IMA J Numer Anal 41(1):638–728
Hipp D, Hochbruck M, Stohrer C (2018) Unified error analysis for non-conforming space discretizations of wave-type equations. IMA J Numer Anal 39(2):1206–1245
Karaa S (2011) Finite element o-schemes for the acoustic wave equation. Adv Appl Math Mech 3(1):181–203
Kashiwabara T, Colciago C, Dedè L, Quarteroni A (2015) Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized robin boundary value problem. SIAM J Numer Anal 53(1):105–126
Kobayashi Y, Tanaka N (2008) An application of semigroups of locally lipschitz operators to carrier equations with acoustic boundary conditions. J Math Anal Appl 338(2):852–872
Komornik V, Zuazua E (1990) A direct method for boundary stabilization of the wave equation. J Math Pure et Appl 69:33–54
Lee MJ, Kang JR (2021) Blow-up results for a quasilinear von karman equation of memory type with acoustic boundary conditions. Appl Math Lett 112
Límaco J, Clark HR, Frota CL, Medeiros LA (2011) On an evolution equation with acoustic boundary conditions. Math Methods Appl Sci 34(16):2047–2059
Lions JL (1969) Quelques méthodes de résolution des problemes aux limites non linéaires, vol 31. Dunod Paris
Medeiros LA, Miranda MM (1996) On a boundary value problem for wave equations: Existence, uniqueness-asymptotic behavior. Revista de Matemáticas Aplicadas, Universidade do Chile 17:47–73
Mugnolo D (2006) Abstract wave equations with acoustic boundary conditions. Math Nachr 279(3):299–318
Shi D, Li Z (2008) Superconvergence analysis of the finite element method for nonlinear hyperbolic equations with nonlinear boundary condition. Appl Math J Chin Univ 23(3):455–462
Silva PB, Clark H, Frota C (2017) On a nonlinear coupled system of thermoelastic type with acoustic boundary conditions. Comput Appl Math 36(1):397–414
Verfürth R (1994) A posteriori error estimation and adaptive mesh-refinement techniques. J Comput Appl Math 50(1):67–83
Vicente A, Frota C (2013) Nonlinear wave equation with weak dissipative term in domains with non-locally reacting boundary. Wave Motion 50(2):162–169
Vicente A, Frota C (2013) On a mixed problem with a nonlinear acoustic boundary condition for a non-locally reacting boundaries. J Math Anal Appl 407(2):328–338
Wheeler MF (1973) \(l_{\infty }\) estimates of optimal orders for galerkin methods for one-dimensional second order parabolic and hyperbolic equations. SIAM J Numer Anal 10(5):908–913
Acknowledgements
Adriano A. Alcântara and Bruno A. Carmo were partially supported by CAPES-Brazil. Mauro A. Rincon was partially supported by CNPq-Brazil.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Abdellah Hadjadj.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Alcântara, A.A., Carmo, B.A., Clark, H.R. et al. Nonlinear wave equation with Dirichlet and Acoustic boundary conditions: theoretical analysis and numerical simulation. Comp. Appl. Math. 41, 141 (2022). https://doi.org/10.1007/s40314-022-01822-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-01822-5
Keywords
- Nonlinear wave equation
- Existence and uniqueness
- Decay of the energy
- Crank–Nicolson Galerkin method
- Convergence order
- Newton’s method