Geodesic
On the discretization of evolution $p$-bi-Laplace equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 4, pp. 371-383.

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This article discusses the mixed finite element method combined with backward-Euler method to study the hyperbolic $p$-bi-Laplace equation, where the existence and uniqueness of solution for discretized problem is shown in Lebesgue Sobolev spaces. The mixed formulation and the inf–sup condition are then given to prove the well posed of the scheme and the optimal a priori error estimates for fully discrete schemes is extracted. Finally, a numerical example is given to confirm the theoretical results obtained. 
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M. Djaghout; A. Chaoui; K. Zennir. On the discretization of evolution $p$-bi-Laplace equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 25 (2022) no. 4, pp. 371-383. http://geodesic.mathdoc.fr/item/SJVM_2022_25_4_a2/

[1] R. Alsaedi, A. Dhifli, A. Ghanmi, “Low perturbations of p-biharmonic equations with competing nonlinearities”, Complex Variables and Elliptic Equations, 66:4 (2021), 642–657

[2] I. Babus̄ka, J. Osborn, J. Pitkäranta, “Analysis of mixed methods using mesh dependent norms”, Math. Comp., 35:152 (1980), 1039–1062

[3] J. H. Bae, J. M. Kim, J. Lee, K. Park, “Existence of nontrivial weak solutions for p-biharmonic Kirchhoff-type equations”, Boundary Value Problems, 125 (2019), 1–17

[4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, 4, North-Holland Publishing Co, Amsterdam, 1978

[5] H. Che, Y. Wang, Z. Zhou, “An optimal error estimates of H1-Galerkin expanded mixed finite element methods for nonlinear viscoelasticity-type equation”, Math. Prob. Eng., 2011, 570980 | DOI

[6] R. E. Ewing, Y. Lin, T. Sun, J. Wang, S. Zhang, “Sharp $L^2$-error estimates and superconvergence of mixed finite element methods for non-fickian flows in porous media”, SIAM J. Numer. Anal., 40:4 (2002), 1538–1560

[7] E. H. Georgoulis, P. Houston, “Discontinuous Galerkin methods for the biharmonic problem”, IMA J. Numer. Anal., 29:3 (2009), 573–594

[8] E. H. Georgoulis, T. Pryer, “Analysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data”, Calcolo, 54:4 (2017), 1533–1551

[9] T. Gyulov, G. Morosanu, “On a class of boundary value problems involving the p-biharmonic operator”, J. Math. Anal. Appl., 367:1 (2010), 43–57

[10] Y. Huang, X. Liu, “Sign-changing solutions for p-biharmonic equations with Hardy potential in the half-space”, J. Math. Anal. Appl., 444 (2016), 1417–1437

[11] A. Lazer, P. McKenna, “Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis”, SIAM Review, 32:4 (1990), 537–578

[12] C. Liu, J. Guo, “Weak solutions for a fourth order degenerate parabolic equation”, Bull. of the Polish Academy of Sciences, Mathematics, 54:1 (2006), 27–39

[13] L. Li, Ch. Tang, “Existence and multiplicity of solutions for a class of p(x)-biharmonic equations”, Acta Mathematica Scientia, 33B:1 (2013), 155–170 | DOI

[14] V. Li, “The $W_p^1$ stability of the Ritz projection on graded meshes”, Math. Comp., 86 (2017), 49–74

[15] C. G. Makridakis, “On the Babus̄ka-Osborn approach to finite element analysis: $L^2$ estimates for unstructured meshes”, Numer. Mathem., 139:4 (2018), 831–844

[16] N. Rajashekar, S. Chaudhary, V. V.K. Srinivas Kumar, “Approximation of p-biharmonic problem using WEB-spline based mesh-free method”, Intern. J. Nonlin. Sci. Numer. Simul., 20:6 (2019), 703–712

[17] D. Sandri, “Sur l'pproximation numérique descoulements quasi-newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau”, RAIRO Model. Math. Anal. Numer., 27:2 (1993), 131–155

[18] Z. Zhou, “On a p(x)-biharmonic problem with Navier boundary condition”, Boundary Value Problems, 149 (2018), 1–14