2020/08/26 by Sana Keita, Keita, Sana, Abdelaziz Beljadid +3
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2008.11879
openalex publication_date 2020/08/26 · openalex created_date 2022/07/22 · openalex updated_date 2026/07/28
We focus here on a class of fourth-order parabolic equations that can be\nwritten as a system of second-order equations by introducing an auxiliary\nvariable. We design a novel second-order fully discrete mixed finite element\nmethod to approximate these equations. In our approach, we propose new\ntechniques using the second-order backward differentiation formula for the time\nderivative and a special technique for the approximation of nonlinear terms.\nThe use of the proposed technique for nonlinear terms makes the developed\nnumerical scheme efficient in terms of computational cost since the proposed\nmethod only deals with a linear system at each time step and no iterative\nresolution is needed. A numerical convergence study is performed using the\nmethod of manufactured and analytical solutions of the system where we\ninvestigate different boundary conditions. With respect to the spatial\ndiscretization, convergence rates are found to at least match a priori error\nestimates available for linear problems. The convergence analysis is completed\nwith an investigation of the temporal discretization where we numerically\ndemonstrate the second-order time-accuracy of the proposed scheme using the\nmethod of reference solution. We present a series of numerical tests to\ndemonstrate the efficiency and robustness of the proposed scheme.\n