NUMERICAL APPROXIMATION OF SOME NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS USING FINITE ELEMENT METHODS

Abstract

This study covers the analysis and computations of some parabolic partial differential equations (PDEs) with the help of finite element methods. Parabolic PDEs contain a time-dependent process, including heat conduction equation, diffusion equation, traffic flow, shallow-water waves, and non-linear acoustics waves. Various analytical methods exist for solving PDEs, including but not limited to the variable separable method, eigenfunction method, Fourier transform, Laplace transform, Green’s function method, and the method of characteristics. However, despite the availability of these methodologies, the scope of our ability to solve PDEs analytically remains limited. While some linear PDEs have known analytic solutions available in the literature, tackling PDEs with non-linear terms presents considerable challenges. Consequently, resorting to advanced simulation techniques to approximate solutions emerges as a preferred approach for model exploration. In this work, we consider some time dependent non-linear PDEs and try to approximate their solutions using conforming finite element methods. We also establish a stability analysis of the proposed method. Soliton-type analytic solutions are also derived via the tanh-coth method. chapter 1 includes the basic concepts regarding finite element methods and a detailed literature survey is presented. In chapter 2, we consider nonlinear advection-diffusion model. The theoretical and numerical aspects of this chapter make up its two main components. As the advection-diffusion model is highly nonlinear and well-posedness is not available in the literature, we need to derive the existence and uniqueness result for it. To do this, we establish the existence and uniqueness of weak formulation using the Picard existence theorem and Banach-Alaoglu theorem in the suitable Sobolev normed spaces. In the second component, we derive convergence results for finite element approximation. This chapter provides the novel finite element error estimates in both L2 and L∞ Bˆochner norms for time-dependent advection-diffusion type problems. The convergence error analysis, which is a major component of this chapter, demands that the stability estimates be derived as well. As addressed in the literature, to derive L2 error estimate, one has to employ suitable projections with certain assumptions to find the error estimates. Though L2 error estimates in elliptic problems are well known, it is not straightforward to employ the same technique for time-dependent problems. For this, we apply the Ritz projection operator without making any assumptions on it and we obtain the error estimates in L2 Bˆochner norms. In chapter 3, the generalized Benjamin-Bona-Mahony-Burgers (GBBMB) model is considered. The main aim of this chapter is to analyze the dynamics of new soliton solutions for the GBBMB model by enforcing the exp-function and tanh-coth methods and numerically validate the obtained analytical solutions via a numerical algorithm based on the Galerkin finite element method. Firstly, we obtained new solitons by applying the tanh-coth and exp function methods to 1D and 2D GBBMB models. After that, for computation purposes, the infinite domain interval is truncated to approximate the finite domain interval. Further, a numerical algorithm is developed to simulate the GBBMB model and to verify the obtained soliton solutions with numerical solutions. To check the accuracy of the reported algorithm, various numerical examples with four different domains are considered. The obtained errors by the developed algorithm are compared with the IEFG method and the Meshless method. In chapter 4, we consider the Rosenau–Burgers model with a fourth-order (biharmonic) dissipative term. In this work, we present a novel and comparative analysis of finite element discretizations for a nonlinear Rosenau–Burgers model, including a biharmonic term. We analyze both continuous and mixed finite element approaches, providing stability, existence, and uniqueness statements of the corresponding variational methods. We also obtain optimal error estimates of the semidiscrete scheme in corresponding Bˆochner spaces. Finally, we construct a fully discrete scheme through a backward Euler discretization of the time derivative and prove well-posedness statements for this fully discrete scheme. Our findings show that the mixed approach removes some theoretical impediments to analysis and is numerically easier to implement. We provide numerical simulations for the mixed formulation approach using C0 Taylor-Hood finite elements on several domains. Our numerical results confirm that the algorithm has optimal convergence in accordance with the observed theoretical results. In chapter 5, the Korteweg-de Vries-Rosenau-regularized long-wave (KdV-RRLW) model is considered. The main objective of this chapter is to analyze and approximate the KdV-RRLW model using the finite element algorithm, focusing specifically on theoretical findings such as the existence and uniqueness of weak solutions. Using the Picard existence theorem, we have demonstrated that the problem has a unique local weak solution for [0, tm], where 0 < tm < T. The continuation principle then implies that there is a global solution in the (0, T) range. The error analysis conducted as a part of this study demanded that we additionally derive certain a priori bounds. For C1 confirming finite element spaces, we obtained optimal H2 and L∞ error estimates using the initial Galerkin approximation as the L2-projection of v0. For the fully discrete scheme, we further derived a priori error estimates based on second order Crank-Nicolson (C-N) scheme. Using the Brouwer fixedpoint theorem, we demonstrated the existence and uniqueness of a fully discrete finite element solution V n. Furthermore, we established the linearized approach to eliminate the difficulties associated with numerically addressing nonlinear problems. Several numerical examples with different domains and initial boundary conditions were considered to evaluate the competence and accuracy of the reported method. In chapter 6, we wish to simulate the two-dimensional KdV-RRLW model based on the Mixed Finite Element method. The theoretical and numerical aspects of this chapter make up its two main components. First, the chapter provides some novel types of error estimates for mixed semi-discrete scheme in different Bˆochner spaces. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with superconvergence order under the H1 norm while the newly introduced variable converges to the exact one with the optimal order under the L2 norm. The model is interesting to analyze as it does not belong to the well-studied classes of time-dependent equations. Hence, we have rigorously proved the wellposedness of weak solutions of the semi-discrete and fully discrete Galerkin mixed finite element method. Furthermore, the fully-discretized problem has been investigated using the Backward-Euler scheme. Secondly, we have verified our theoretical findings on various domains using C0 finite elements associated with piecewise polynomials of degree 2 (P2 element) for several numerical experiments in one and two-dimensional cases. Here, results are provided for non-smooth initial data as well. Further, we have provided a comparison of the proposed method with other methods. This is the first instance where a numerical scheme is developed for a nonlinear biharmonic problem with all the positive coefficients α, β, γ and λ. In chapter 7, we wish to simulate a new multidimensional electrolyte model in mechanical equilibrium by employing the finite element method. The main focus of this chapter is to provide a novel and structured framework for simulating a new multidimensional electrolyte model by employing the confirming finite element approximation. Till date, this aspect has not been addressed in the literature, and the current study is the first attempt to employ confirming finite element approximation for simulating the new electrolyte model across such diverse spatial dimensions. This work is characterized by its comprehensive consideration of both compressible and incompressible variants of the electrolyte mixture. Furthermore, the finite ion size effects, which play a significant role in the dynamics of electrolyte solutions, are also considered in this study. By doing this, we improve the precision and robustness of our numerical simulations, guaranteeing a more robust handling of low concentrations compared to alternative methods. chapter 8 includes the Conclusions and Future Scopes.