Streamline Diffusion Finite Element Method Research Articles

SDFEM for Time-Dependent Singularly Perturbed Boundary Turning Point Problem

A time-dependent singularly perturbed boundary turning point problem is considered. The numerical approximation to the solution of the problem is achieved by using a weighted-[Formula: see text] method on uniform mesh in time direction and streamline diffusion finite element method (SDFEM) on layer adapted mesh in space direction. Properties of the discrete Green’s functions are utilized to ensure [Formula: see text]-uniform stability and optimality of the SDFEM. We perform the convergence analysis for the proposed scheme both in space and time direction. The convergence results are obtained in the maximum norm by the decomposition of the solution into smooth and layer component. It is demonstrated that the proposed numerical scheme is [Formula: see text]-uniformly convergent of order one for ([Formula: see text]) and of almost order two for ([Formula: see text]). Some relevant numerical examples are taken to illustrate the theoretical findings and efficiency of the numerical method.

Streamline diffusion finite element method for a singularly perturbed problem with an interior layer on Shishkin mesh

With the continuous development of science and technology, singularly perturbed problems have attracted more and more attentions from computational fluid scholars. In this paper, we discuss a singularly perturbed convection–diffusion problem with a discontinuous convection. Generally speaking, due to this discontinuity, an interior layer appears in the solution. A streamline diffusion finite element method on Shishkin mesh is used to solve this problem, and the optimal order of convergence in a modified streamline diffusion norm is derived. Numerical results are presented to support the theoretical conclusion.

Streamline Diffusion Finite Element Method for Singularly Perturbed 1D-Parabolic Convection Diffusion Differential Equations with Line Discontinuous Source

This article presents a study on singularly perturbed 1D parabolic Dirichlet’s type differential equations with discontinuous source terms on an interior line. The time derivative is discretized using the Euler backward method, followed by the application of the streamline–diffusion finite element method (SDFEM) to solve locally one-dimensional stationary problems on a Shishkin mesh. Our proposed method is shown to achieve first-order convergence in time and second-order convergence in space. Our proposed method offers several advantages over existing techniques, including more accurate approximations of the solution on the boundary layer region, better efficiency, and robustness in dealing with discontinuous line source terms. The numerical examples presented in this paper demonstrate the effectiveness and efficiency of our method, which has practical applications in various fields, such as engineering and applied mathematics. Overall, our proposed method provides an effective and efficient solution to the challenging problem of solving singularly perturbed parabolic differential equations with discontinuous line source terms, making it a valuable tool for researchers and practitioners in various domains.

Supercloseness of linear streamline diffusion finite element method on Bakhvalov-type mesh for singularly perturbed convection-diffusion equation in 1D

For the singularly perturbed convection-diffusion equations, the uniform convergence of the finite element method on Bakhvalov-type mesh is still in its infancy. The challenge is how to handle the analysis on the last element mesh in the layer. In this manuscript, a streamline diffusion finite element method is properly defined on a Bakhvalov-type mesh. By means of delicate analysis of the terms on the last element mesh in the layer, we derive the supercloseness of almost order 2, which is consistent with the numerical experiments.

Convergence analysis of a fully-discrete FEM for singularly perturbed two-parameter parabolic PDE

This article deals with the numerical solution of a singularly perturbed initial–boundary value problem (IBVP) with two parameters on a rectangular space–time domain. A fully-discrete numerical method by combining the Crank–Nicolson scheme for temporal derivative and the streamline-diffusion finite element method (SDFEM) for spatial derivatives has been established in a new theoretical framework. A suitable stabilization parameter has been explored on some conditions related to error analysis. The robust error estimates and the stability results are obtained by using P1-finite element in the discrete L2(0,T;SD)-norm. Theoretical results are verified by some numerical experiments.

Stability and error analysis of a fully-discrete numerical method for system of 2D singularly perturbed parabolic PDEs

This article deals with a fully-discrete numerical method for the system of 2D singularly perturbed parabolic convection-diffusion problems. The method is based on the combination of the implicit Backward-Euler scheme for the temporal derivative and the streamline-diffusion finite element method (SDFEM) for the spatial derivatives. Due to the existence of exponential boundary layers in the solution of this problem, a piecewise-uniform layer-adapted mesh is used here for the triangulation of the spatial domain. The stability of the present method has been discussed based on an appropriate stabilization parameter. The convergency of the fully-discrete SDFEM on a discrete L2(0,T;L2)-norm using the piecewise continuous linear finite element space is analyzed in a new theoretical framework. Finally, the justification of the theoretical results is done by some numerical experiments.

Asymptotic streamline diffusion finite element method for singularly perturbed convection–diffusion differential difference equations

In this article, we presented an asymptotic streamline diffusion finite element method (SDFEM) for singularly perturbed convection–diffusion‐type differential difference equations. Using the asymptotic expansion approximation, the differential difference equations are converted into nondelay differential equations. A SDFEM is applied to the modified equation. Further, we proved that the method gives an almost second‐order convergence in maximum norm, in L2‐norm whereas first‐order convergence in H1‐norm. Numerical results are presented to validate the theoretical results.

Numerical analysis of a fully discrete stabilized FEM for system of singularly perturbed parabolic IBVPs

In this article, a fully discrete numerical method is studied to solve the system of singularly perturbed parabolic convection–diffusion problem. The full discretization incorporates the backward-Euler method and the streamline-diffusion finite element method (SDFEM) for the temporal and spatial discretization, respectively. In each equation, the highest ordered spatial derivatives are multiplied by distinct parameters, hence the overlapping boundary layers phenomena displayed in the solution. To capture the singularity of the solution, a layer-adapted mesh has been used for the spatial domain discretization. The stability of the numerical method has been addressed by considering an appropriate stabilization parameter. A complete error analysis has shown the first order of accuracy of the fully discrete method in the discrete -norm. Lastly, few numerical experiments have been addressed to justify the theoretical estimates.

An asymptotic streamline diffusion finite element method for singularly perturbed convection‐diffusion delay differential equations with point source

In this article, we presented an asymptotic SDFEM for singularly perturbed convection diffusion type differential difference equations with point source term. First, the solution is decomposed into two functions, among them one is the solution of delay differential equation and other one is the solution of differential equation with point source. Furthermore, using the asymptotic expansion approximation, the delay differential equation is modified as a nondelay differential equations. Streamline diffusion finite element methods are applied to approximate the solutions of the two problems. We prove that the present method gives an almost second-order convergence in maximum norm and square integrable norm, whereas first-order convergence in H 1 norm. Numerical results are presented to validate the theoretical results.

A novel two-step streamline-diffusion FEM for singularly perturbed 2D parabolic PDEs

In this article, a class of singularly perturbed 2D parabolic convection-diffusion-reaction initial-boundary-value problem is numerically solved by the alternating direction implicit type operator splitting streamline-diffusion finite element method. The proposed scheme alleviates the computational complexity and the high storage requirement for higher-dimensional problems. The overall stability of the two-step method is established. A piecewise-uniform Shishkin mesh is used for the spatial domain discretization. ε-uniform error estimate has been derived with an appropriate choice of the stabilization parameter. The parameter is shown to depend on the time step-length which is necessary for the stability of the method. Some numerical simulations are carried out to validate the theoretical error estimate.

A SDFEM for Singularly Perturbed System of Convection-Diffusion Delay Differential Equations

In this article, we consider a system of convection-diffusion equations with delay terms. When a parameter multiplying the second order derivatives in the equations is small, boundary layers as well as interior layers appear in their solutions. A numerical method based on finite element and Shishkin mesh is presented. We derive an error estimate of order O(N −1 log2 N) in the energy norm with respect to the parameter using the streamline-diffusion finite element method. Numerical experiments are also presented to support our theoretical results.

SDFEM for singularly perturbed parabolic initial-boundary-value problems on equidistributed grids

In this article, we study the convergence properties of the streamline-diffusion finite element method (SDFEM) for singularly perturbed 1D parabolic convection–diffusion initial-boundary-value problems. To discretize the spatial domain, we use a layer-adaptive nonuniform grids obtained through the equidistribution principle, whereas uniform grid is used in the time direction. Here, we use the backward-Euler method to discretize the temporal derivative and the SDFEM scheme for the spatial derivatives. The proposed method is uniformly convergent with first-order in time and second-order in space.

SDFEM for singularly perturbed boundary-value problems with two parameters

In this article, we study the convergence of the streamline-diffusion finite element method (SDFEM) for singularly perturbed boundary-value problem with two parameters. We prove that the SDFEM is uniformly convergent in the discrete SD-norm, of second-order on the Shishkin mesh and almost second-order on the Duran–Shishkin mesh. Numerical results are presented to support the theoretical error estimates.

A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces

We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included.

Optimal adaptivity for the SUPG finite element method

For convection dominated problems, the streamline upwind Petrov–Galerkin method (SUPG), also named streamline diffusion finite element method (SDFEM), ensures a stable finite element solution even on coarse meshes, i.e., in the preasymptotic range. Based on some a posteriori error estimators from the literature, we formulate an adaptive mesh-refining algorithm for SUPG. We prove that the adaptively generated SUPG solutions converge at asymptotically optimal rates towards the exact solution. The main focus thus is on the mathematical proof that the SUPG stabilization does not spoil the asymptotic convergence behavior of the adaptive scheme.

The streamline-diffusion finite element method on graded meshes for a convection–diffusion problem

In this paper, the streamline-diffusion finite element method is applied to a two-dimensional convection–diffusion problem posed on the unit square, using a graded mesh of O(N2) points based on standard Lagrange polynomials of degree k≥1. We prove the method is convergent almost uniformly in the perturbation parameter ϵ, and a convergence order O(N−klogk+1⁡(1ϵ)) is obtained in a streamline-diffusion norm under certain assumptions. Numerical experiments support the theoretical results.

SDFEM for an elliptic singularly perturbed problem with two parameters

A singularly perturbed problem with two small parameters in two dimensions is investigated. Using its discretization by a streamline-diffusion finite element method with piecewise bilinear elements on a Shishkin mesh, we analyze the superconvergence property of the method and suggest the choice of stabilization parameters to attain optimal error estimate in the corresponding streamline-diffusion norm. Numerical tests confirm our theoretical results.

Streamline diffusion finite element method for stationary incompressible natural convection problem

In this article, a streamline diffusion finite element method is proposed and analyzed for stationary incompressible natural convection problem (NC). This method is stable for any combinations of velocity, pressure, and temperature finite element spaces, without requiring inf-sup condition. The well-posedness and convergence (at optimal error rate) of this scheme are proved in terms of some conditions. Numerical experiments are illustrated to validate our theoretical analysis and show the streamline diffusion finite element approach is effective for solving the NC problems.

Analysis of the SDFEM for singularly perturbed differential–difference equations

In this paper, the stability and accuracy of a streamline diffusion finite element method (SDFEM) for the singularly perturbed differential–difference equation of convection term with a small shift is considered. With a special choice of the stabilization quadratic bubble function and by using the discrete Green’s function, the new method is shown to have an optimal second order in the sense that $$\Vert u-u_{h}\Vert _{\infty }\le C\inf \nolimits _{v_h\in V^h}\Vert u-v_{h}\Vert _{\infty }$$ , where $$u_{h}$$ is the SDFEM approximation of the exact solution u in linear finite element space $$V_{h}$$ . At last, a second order uniform convergence result for the SDFEM is obtained. Numerical results are given to confirm the $$\varepsilon $$ -uniform convergence rate of the nodal errors.

A Priori Error Estimates and Computational Studies for a Fermi Pencil-Beam Equation

We derive a priori error estimates for the standard Galerkin and streamline diffusion finite element methods for the Fermi pencil-beam equation obtained from a fully three-dimensional Fokker–Planck equation in space and velocity variables. For a constant transport cross-section, there is a closed form analytic solution available for the Fermi equation with a data as product of Dirac functions. Our objective is to study the case of nonconstant, nonincreasing transport cross-section. Therefore we start with a theoretical, that is, a priori, error analysis for a Fermi model with modified initial data in L2. Then we construct semi-streamline-diffusion and characteristic streamline-diffusion schemes and consider an adaptive algorithm for local mesh refinements. To derive the stability estimates, for simplicity, we rely on the assumption of nonincreasing transport cross-section. Different numerical examples, in two space dimensions are justifying the theoretical results. Implementations show significant reduction of the computational error by using such adaptive procedure.