On Mesh-free Method for Singularly Perturbed Problems

Abstract

In the present thesis, an attempt has been made to demonstrate some robust and efficient mesh-free techniques for approximating the solutions of singularly perturbed problems. It is observed that the proposed mesh-free method are more efficient than the conventional methods and are, at the same time, conceptually simple. We consider one and two-dimensional singularly perturbed problems commonly arise in the field of science and engineering. Besides it, a well-developed theory for their convergence and error estimates has also been proved. Numerical experiments have been carried out extensively to support the theoretical results. The research work started with the following three objectives: • To propose and implement a mesh-free method for solving linear/non-linear singularly perturbed problems in one dimension. • To implement the proposed mesh-free method for solving the higher dimensional singularly perturbed problem. • To implement the proposed mesh-free method for solving a real-life problem. To accomplish these objectives, the research work has been carried out and organized in form of Chapters. The thesis is categorized into seven Chapters. A brief overview of the Chapters is given below: In Chapter 1 of the proposed thesis work focuses on singularly perturbed differential equations and their importance in various fields. It provides an introduction to basic definitions and notations related to singularly perturbed problems (SPP), highlighting their significance in real-world applications. The Chapter emphasizes on the need for efficient and accurate methods for solving SPP, and reviews various analytic and numerical techniques proposed in this regard. The discussion also includes a summary of meshfree schemes and their developments, highlighting their advantages and limitations. The presented literature survey on this topic has led to the identification of gaps in the existing approaches which motivates for the objectives of the Ph.D. thesis. Overall, Chapter 1 lays the foundation for the subsequent Chapters by providing a clear and concise overview of the problem domain. Chapter 2 deals with one-dimensional singularly perturbed linear and non-linear elliptic differential equations. Singularly perturbed problems have solutions that exhibit sharp boundary layers, which makes them difficult to solve using traditional numerical methods. This Chapter discusses the element-free Galerkin (EFG) methodology for solving the SPP. Due to the absence of element connectivity, the EFG method’s main feature lies in its v ability to add or delete node particles without much difficulty. This feature makes the method more adaptable than other conventional numerical techniques. The EFG method uses moving least-square (MLS) approximation to generate shape functions. The impact of different weight functions on the accuracy of the method has been discussed. The methodology is based on the global weak form, and the numerical integrations are computed using background cells created by the Gauss quadrature formula. Since the MLS shape functions do not satisfy the Kronecker delta function property, the boundary conditions can not be implemented directly. Therefore, Lagrange multipliers approach has been used to impose the essential boundary constraints. The quasilinearization technique has been adopted for handling the nonlinearity present in the considered problems and its rate of convergence has also been derived. Shishkin’s approach has been utilized to generate more nodes in the boundary layer region and to capture these layers sharply. The EFG method’s robustness is verified through various numerical examples and L∞−errors have been presented. Comparisons of solutions have been made with those available in the literature. Chapter 3 of the thesis focuses on the numerical analyzation of the time-dependent singularly perturbed parabolic reaction-diffusion equation. To accomplish this, the authors utilized the element-free Galerkin (EFG) method for spatial discretization in conjunction with the implicit Crank-Nicolson scheme for temporal discretization. Due to the steep boundary layers in the solution to the problem, the authors employ a piecewise uniform layer-adapted Shishkin’s technique to generate more dense node points near the boundary layer region. Stability and a posteriori errors of the proposed method have been analyzed using L2-norm. The -uniform convergence of the full-discrete scheme is shown to be O(τ 2+ds m), where τ represents the time step size and ds represents the size of the influence domain. Some numerical experiments have been conducted to validate the theoretical results and to verify the computational consistency and robustness of the proposed scheme. The numerical order of convergence has also been presented. Chapter 4 is dedicated to demonstrate the application of the proposed method for solving real-life problems. In particular, the study focuses on a realistic model that displays the phenomenon of traveling wave propagation. The time-dependent singularly perturbed Fisher’s problem, represented by the below given equations, has been used as the model problem for analysis: ut(x, t) = ∇2u(x, t) + βu(x, t)(1 − u(x, t)), (x, t) ∈ Ω ≡ Ωx × Ωt ≡ (0, 1) × (0, T], with initial condition u(x, 0) = u0(x), x ∈ Ω¯ x, vi and boundary conditions as u(0, t) = f(t), u(1, t) = g(t), t ∈ Ω¯ t , where u(x, t) symbolizes the occurrence of traveling waves. Implicit Crank-Nicolson technique has been employed for temporal semi-discretization. Spatial discretization has been carried out using mesh-free element-free Galerkin scheme. The authors analyzed the stability of the semi-discrete scheme. The non-linear terms has been tackled by using the quasilinearization process. The convergence analysis of the quasilinearization process and the full discrete scheme has also been discussed. Numerical experiments have been performed to validate the theoretical findings and to show the robustness of the proposed scheme. The numerical results demonstrate the efficiency and accuracy of the proposed method in solving the time-dependent singularly perturbed Fisher’s problem. Chapter 5, a more generalized version of the Fisher’s model problem, i.e. Burger-Fisher’s problem, has been considered to check the robustness of the proposed EFG method. The considered Burger-Fisher’s model problem is given by ∂u ∂t − γ ∂u2 ∂ 2x + αuδ ∂u ∂x = F(u), (x, t) ∈ Ω ≡ Ωx × Ωt ≡ (0, 1) × (0, T], u(x, 0) = u0(x), x ∈ Ω¯ x, u(0, t) = f(t), u(1, t) = g(t), t ∈ Ω¯ t . Mesh-free EFG method along with Crank-Nicolson scheme has been proposed to analyze the realistic model problem which generally arises in the biological field. The shape functions are generated using the moving least-square (MLS) approximation and the Lagrange multiplier method has been invoked to implement the essential boundary conditions. The existence and uniqueness of the EFG solution has been presented. Stability and uniform convergence of the proposed scheme for the non-linear model problem has also been presented for fixed value of singular perturbation parameter. Numerical results have been presented which depicts the efficiency of the scheme. In Chapter 6, the proposed scheme has been extended to solve the two-dimensional time-dependent non-linear singularly perturbed reaction-diffusion initial-boundary value problem. The temporal and spatial discretizations have been carried out using the CrankNicolson and element-free Galerkin scheme respectively. Again the MLS approach has been invoked to generate the basis functions. To impose the boundary conditions, Lagrange multiplier method has been utilized. Stability of the time semi-discrete problem has been vii analyzed. Uniform convergence of the proposed scheme has been shown in the L2− norm. Numerical results have been presented to validate the efficiency of the method. In Chapter 7, we have investigated a real-life application of a two-dimensional singularly perturbed parabolic equation in the context of finance. Specifically, the Black-Scholes (B-S) model has been considered, commonly used for option pricing and based on the principle of hedging to eliminate risks associated with underlying assets and stock options. The two-dimensional Black-Scholes (B-S) equation for a European call option pricing is given by: ∂C ∂t (S1, S2, t) + 1 2 σ 2 1S 2 1 ∂ 2C ∂S2 1 (S1, S2, t) + 2ρσ1σ2S1S2 ∂ 2C ∂S1∂S2 (S1, S2, t) + 1 2 σ 2 2S 2 2 ∂ 2C ∂S2 2 (S1, S2, t) + r h S1 ∂C ∂S1 (S1, S2, t) + S2 ∂C ∂S2 (S1, S2, t) i − rC(S1, S2, t) = 0, (S1, S2, t) ∈ (0,∞) × (0, ∞) × (0, T), with initial condition C(S1, S2, 0) = S2 max(S1 − E, 0), (S1, S2) ∈ (0,∞) × (0, ∞) and terminal conditions C(0, S2, t) = 0 = C(S1, 0, t), t ∈ [0, T] where C(S1, S2, t) denotes the value of a call option with underlying asset prices S1, S2 at time 0 t 0 . Here, 0σ 0 1 and 0σ 0 2 are the volatility rates of the underlying assets S1 and S2, respectively. 0 r 0 is the risk-free interest rate, 0ρ 0 is correlation value between S1, S2 and E is the strike or exercise price. A dimensionless interpretation of the two-dimensional B-S equation has been demonstrated by using linear transformations. The above model has been reformulated into initial-value singularly perturbed parabolic problem as follows: LU(x, y, τ ) ≡ ( − ∂ ∂τ + 1 ∂ 2 ∂x2 + 2 ∂ 2 ∂y2 − 1 ) U(x, y, τ ) = 0. Here, 1 = σ 2 1 2r , 2 = σ 2 2 2r are dimensionless perturbation parameters, 1, 2 ∈ (0, 1]. The numerical treatment of the deemed problem has been accomplished by adopting the element-free Galerkin (EFG) approach. The proposed mesh-free scheme is novel for the singularly perturbed B-S equation for 0 < 1, 2 << 1. An implicit Crank-Nicolson scheme has been used to discretize temporal variable. Error analysis of the scheme reveals an order viii of convergence O(τ 2+ds m), where τ and ds m are the time step size and size of the influence domain, respectively. Numerical experiments verify the theoretical and computational consistency of the proposed scheme, and L∞−errors of solutions with sharp boundary layers are presented. Conclusion of the thesis has been presented in the last followed by bibliography.