Bakhvalov Mesh Research Articles

Supercloseness of linear finite element method on Vulanović–Bakhvalov mesh for singularly perturbed convection–diffusion equation in 1D

Supercloseness of linear finite element method is studied on the Vulanović–Bakhvalov mesh for a singularly perturbed convection–diffusion problem. A novel interpolation is proposed to tackle challenges arising from the subinterval adjacent to the transition point. This innovative interpolation method is formulated by leveraging the distinctive features of the boundary layer and incorporating the specific attributes of the Vulanović–Bakhvalov mesh. On the basis of that, supercloseness of 2th order can be obtained.

Supercloseness analysis of the linear finite element method for a singularly perturbed convection–diffusion problem on Vulanović–Bakhvalov mesh

Vulanović–Bakhvalov mesh is a kind of Bakhvalov-type mesh, which has the advantages of good numerical convergence and the transition point between coarse and fine mesh is not affected by mesh parameters. However, there is little theory of finite element methods for this mesh. In this paper, a linear finite element method on a Vulanović–Bakhvalov mesh is studied for a singularly perturbed convection–diffusion problem. In this process, some properties of the Vulanović–Bakhvalov mesh are obtained. Based on these properties and a new Lagrange interpolation, we further derive the supercloseness result of 3/2th order in an energy norm.

Richardson extrapolation method for 2D-SPP on VB mesh singularly perturbed convection–diffusion problem on a Vulanović–Bakhvalov mesh

Richardson extrapolation method for 2D-SPP on VB mesh singularly perturbed convection–diffusion problem on a Vulanović–Bakhvalov mesh

Analysis of Numerical Differential Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer

The paper considers numerical differentiation of functions with large gradients in the region of an exponential boundary layer. This topic is important, since the application of classical polynomial difference formulas for derivatives to such functions in the case of a uniform mesh leads to unacceptable errors if the perturbation parameter is comparable with the mesh size. The numerical differentiation formula with a given number of nodes in the difference stencil is built on subintervals covering the original interval. The accuracy of numerical differentiation formulas on a Bakhvalov mesh, which is widely used in the construction of difference schemes for singularly perturbed problems, is analyzed. For the original function of one variable, a representation in the form of a sum of regular and boundary-layer components, based on the Shishkin decomposition, is used to solve a singularly perturbed problem. Previously, such a decomposition was used to prove the convergence of difference schemes. An estimate of the error of classical polynomial formulas for numerical differentiation on a Bakhvalov mesh is obtained. The error estimate on a Bakhvalov mesh is obtained in the general case, when a derivative of an arbitrarily given order is calculated and the difference stencil for this derivative contains a given number of nodes. The error estimate depends on the order of the calculated derivative and the number of nodes in difference stencil and takes into account the uniformity in the parameter . The results of numerical experiments are presented, which are consistent with the error estimates obtained.

Analysis of Numerical Differential Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer

Analysis of Numerical Differential Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer

A parameter uniform method for two-parameter singularly perturbed boundary value problems with discontinuous data

We consider two-parameter singularly perturbed problems of reaction-convection-diffusion type in one dimension. The convection coefficient and source term are discontinuous at a point in the domain. The problem is numerically solved using the upwind difference method on an appropriately defined Shishkin-Bakhvalov mesh. At the point of discontinuity, a three-point difference scheme is used. A convergence analysis is given and the method is shown to be first-order uniformly convergent with respect to the perturbation parameters. The numerical results presented in the paper confirm our theoretical results of first-order convergence. Summing up:• The Shishkin-Bakhvalov mesh is graded in the layer region and uniform in the outer region as shown in the graphical abstract.• The method presented here has uniform convergence of order one in the supremum norm.• The numerical orders of convergence obtained in numerical examples with Shishkin- Bakhvalov mesh are better than those for Shishkin mesh.

High-order finite element method on a Vulanović–Bakhvalov mesh for a singularly perturbed convection–diffusion problem

A kth order (k≥1) finite element method on a Vulanović–Bakhvalov mesh is proposed for a singularly perturbed convection–diffusion problem. Some properties of the Vulanović–Bakhvalov mesh are derived. Based on these estimations and a new Lagrange interpolation technique, an optimal order of uniform convergence with respect to the perturbation parameter is proved. Numerical results are provided to demonstrate the theoretical findings.

Analysis of a Fitted Finite Difference Scheme for a Semilinear System of Singularly Perturbed Reaction-Diffusion Equations Having Discontinuous Source Term

In this paper, a system of an arbitrary number of singularly perturbed semilinear differential equations of reaction-diffusion type having discontinuous source term is examined, for the case in which the diffusion parameters associated with each equation of the considered system are assumed to be different in magnitude. In addition to the occurrence of the boundary layers, the solution exhibits the overlapping and interacting internal layers due to the existence of discontinuity in the data. A central finite difference scheme is used in conjunction with a suitable piecewise uniform Shishkin mesh and a Bakhvalov mesh, to construct the numerical method. Using discrete Green’s function technique, the proposed numerical scheme has been proved to be an almost second-order of uniform convergent for the Shishkin mesh and second-order of uniform convergent for the Bakhvalov mesh, independent of the perturbation parameters. Numerical test examples are presented to demonstrate the performance of the numerical scheme.

A new upwind difference analysis of an exponentially graded Bakhvalov-type mesh for singularly perturbed elliptic convection–diffusion problems

We are concerned with numerical methods for singularly perturbed elliptic convection–diffusion differential equations in two dimensions. Due to the presence of exponential boundary layers in these problems, special meshes – which are dense in layer regions – are designed to robustly resolve the layers. Recently, the so-called exponentially graded mesh has been extensively studied by Xenophontos et al. (see, e.g., (Xenophontos et al., 2016)) for which the authors classify the mesh as a Shishkin-type mesh. In this paper, we propose a modification of the exponentially graded mesh as a Bakhvalov-type mesh. The modified mesh is constructed by incorporating Bakhvalov’s idea that allows the mesh not only to be gradually graded in the layer regions, but also to preserve the favorable properties of the original Bakhvalov mesh. Our numerical experiments show an improvement in the accuracy of the computed solution when an upwind difference scheme is discretized on our modified mesh. More importantly, the main purpose of the paper is to propose a novel finite-difference error analysis, which is based on the truncation error and barrier function approach, for a class of Bakhvalov-type meshes; that is although such analysis for two-dimensional singularly perturbed convection–diffusion problems on Shishkin-type meshes was derived two decades ago, but no similar results are known for Bakhvalov-type meshes (Roos and Stynes, 2015).

A second-order difference scheme for the singularly perturbed Sobolev problems with third type boundary conditions on Bakhvalov mesh

We consider a singularly perturbed initial-third boundary value Sobolev problem. Firstly, the asymptotic behaviour of the exact solution is analysed. Then, a second-order finite difference scheme is constructed on the special non-uniform mesh. By using energy estimate, the stability and convergence of the proposed scheme are investigated in the discrete energy norm. Finally, three numerical examples are solved to validate the theory.

Analysis of approaches to spline interpolation of functions with large gradients in the boundary layer

The problem of cubic spline interpolation of functions with large gradients in the boundary layer is considered. It is assumed that the decomposition in the form of the sum of the regular and singular components is valid for the interpolated function. This decomposition is valid for the solution of a boundary value problem for a second order ordinary differential equation with a small parameter e at the highest derivative. An overview of approaches to the construction of splines, the error of which is uniform with respect to the small parameter e, is given. The approaches are based on the use of Shishkin and Bakhvalov meshes, which are thickened in the boundary layer region. An alternative approach based on a modification of a cubic spline with fitting to the singular component of the interpolated function is also considered. The comparison of the accuracy of the applied approaches with the performance of computational experiments is carried out.

Lagrange Interpolation and the Newton–Cotes Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer

Lagrange Interpolation and the Newton–Cotes Formulas on a Bakhvalov Mesh in the Presence of a Boundary Layer

Application of Cubic Splines on Bakhvalov Meshes in the Case of a Boundary Layer

Application of Cubic Splines on Bakhvalov Meshes in the Case of a Boundary Layer

An Analysis of the Robust Convergent Method for a Singularly Perturbed Linear System of Reaction–Diffusion Type Having Nonsmooth Data

In this paper, a coupled system of [Formula: see text] second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.

New approaches to constructing quadrature formulas for functions with large gradients

The problem of numerical integration of functions with large gradients in the boundary layer is studied. The use of the Newton-Cotes formulas on a uniform grid to integrate such functions can lead to significant errors. The article investigates approaches to the construction of quadrature formulas, the error of which does not increase due to large gradients of the function in the boundary layer. The following approaches are considered: the construction of quadrature formulas that are exact on the singular component responsible for the growth of the function in the boundary layer, the application of the Newton-Cotes formulas on the Shishkin and Bakhvalov meshes, minimization the error of the quadrature formula due to the construction of nodes. The investigated approaches are new, partially reflected in the publications. The results of computational experiments are presented.

Application a cubic spline to calculate derivatives in the presence of a boundary layer

The problem of calculating the derivatives of a function with large gradients given at the grid nodes is considered. The decomposition of the function into regular and singular components is assumed. Such decomposition takes place in the presence of an exponential boundary layer. The case of the Bakhvalov mesh is considered. The error in calculating the derivatives based on the differentiation of the interpolating cubic spline is estimated. Error estimates are obtained taking into account uniformity in a small parameter.

The cubic interpolation spline for functions with boundary layer on a Bakhvalov mesh

The problem of cubic spline interpolation on the Bakhvalov mesh of functions with region of large gradients is considered. Asymptotically accurate error estimates O(N −4) are obtained for a class of functions with an exponential boundary layer in case 1/N ≤ ε, where N is number of nodes, ε is small parameter. In case ε ≤ 1/N we have experimentally shown that the error estimates of traditional spline interpolation are not uniform in a small parameter, and the error itself can increase indefinitely when the small parameter tends to zero at a fixed number of nodes N. A modified cubic spline is proposed for which uniform estimates of the order O(N −4) have been experimentally confirmed.

High‐order convergent methods for singularly perturbed quasilinear problems with integral boundary conditions

In this work, we develop a numerical scheme for a class of singularly perturbed quasilinear problems with integral boundary conditions. The quasilinear equation is discretized using a hybrid scheme, and the composite trapezoidal rule is used to discretize the boundary condition. We construct a general error analysis framework for the discrete scheme. Within this framework, the discrete scheme is shown to be uniformly convergent of on Shishkin meshes and on Bakhvalov meshes. Further, we propose adaptive generation of meshes based on a suitable monitor function and the mesh equidistribution principle. We prove that on these meshes the discrete scheme is uniformly convergent of Our theoretical findings are supported by numerical results obtained through experiments.

The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems

A linear two-dimensional singularly perturbed convection-diffusion boundary-value problem is considered. The problem is discretized by the upwind finite-difference method. The analysis of this method on Shishkin-type meshes has been well-established, but the discretization mesh in this paper is the original Bakhvalov mesh, introduced in 1969 as the first layer-adapted mesh. We analyze the error of the numerical solution in the maximum norm and prove first-order pointwise accuracy, uniform in the perturbation parameter. This is the first complete analysis of this kind for two-dimensional convection-diffusion problems discretized on the Bakhvalov mesh. Our numerical experiments validate the theoretical analysis.

Numerical differentiation on the Bakhvalov mesh in the presence of an exponential boundary layer

The question of the application of formulas of the numerical differentiation of functions in the presence of the exponential boundary layer is investigated. The problem is that the application of classical formulas, wich are based on the differentiation of the Lagrange polynomial on the uniform mesh in this case leads to significant errors. It is proposed to study the formulas for derivatives on the Bakhvalov mesh, which is widely used in the construction of difference schemes for singularly perturbed problems. It is proved that applying of classical difference formulas for derivatives on a Bakhvalov mesh have error estimate that is uniform with respect to a small parameter. The results of numerical experiments are presented.