Mesh Grading Research Articles

Optimally accurate second-order numerical scheme on layer-resolving graded mesh for singularly-perturbed boundary turning point problems exhibiting power-of-type-1 and hybrid layers

The paper discusses singularly-perturbed problems with boundary turning points. It describes constructions of layer-resolving grids based on layer-eliminating coordinate transformations and analyses the convergence of numerical solutions obtained using the upwind scheme on the layer-resolving grids and Richardson extrapolations. Theoretical error estimates and numerical experiments confirm that the upwind scheme with Richardson extrapolation on the proposed layer-resolving grids is uniformly convergent of order two without any logarithmic factor.

A Finite Element Analysis in Balanced Norms for a Coupled System of Singularly Perturbed Reaction-Diffusion Equations

Abstract In this paper we study approximations of a singularly perturbed system of two coupled reaction-diffusion equations, in one dimension, by using piecewise linear finite elements on graded meshes. When the parameters are of different magnitudes, the solution exhibits in general two distinct but overlapping boundary layers. We prove that, when the mesh grading parameter is appropriately chosen, optimal error estimates in a balanced norm for piecewise linear elements can be obtained. Supporting numerical results are also presented.

Higher order numerical approximations for non-linear time-fractional reaction–diffusion equations exhibiting weak initial singularity

In the present study, we introduce a high-order non-polynomial spline method designed for non-linear time-fractional reaction–diffusion equations with an initial singularity. The method utilizes the L2-1σ scheme on a graded mesh to approximate the Caputo fractional derivative and employs a parametric quintic spline for discretizing the spatial variable. Our approach successfully tackles the impact of the singularity. The obtained non-linear system of equations is solved using an iterative algorithm. We provide the solvability of the novel non-polynomial scheme and prove its stability utilizing the discrete energy method. Moreover, the convergence of the proposed scheme has been established using the discrete energy method in the L2 norm. It is proven that the method is convergent of order min{θα,2} in the temporal direction and 4.5 in the spatial direction, where α∈(0,1) denotes the order of the fractional derivative and the parameter θ is utilized in the construction of the graded mesh. Finally, we conduct numerical experiments to validate our theoretical findings and to illustrate how the mesh grading influences the convergence order when dealing with a non-smooth solution to the problem.

Non-coercive Neumann Boundary Control Problems

The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both the state and adjoint state equations. The article establishes the existence and uniqueness of solutions for both equations, with minimal assumptions made about the problem’s data. Next, the regularity of these solutions is studied in three frameworks: Hilbert–Sobolev spaces, Sobolev–Slobodeckiĭ spaces, and weighted Sobolev spaces. These regularity results enable a numerical analysis of the finite element approximation of both the state and adjoint state equations. The results cover both convex and non-convex domains and quasi-uniform and graded meshes. Finally, the optimal control problem is analyzed and discretized. Existence and uniqueness of the solution, first-order optimality conditions, and error estimates for the finite element approximation of the control are obtained. Numerical experiments confirming these results are included.

L1-FEM discretizations for two-dimensional multiterm fractional delay diffusion equations

A two-dimensional multiterm fractional delay diffusion equation is considered. The representation of the exact solution of the equation is derived and it is shown that the solution exhibits singular behaviors at multiple nodes due to the initial singularity and time delay. This results in the numerical schemes for solving the equation typically have a lower order of convergence in time. The problem is approximated in time by the L1 and Alikhanov schemes on symmetrical graded meshes, while in space the standard finite element method is applied. Numerical stability and convergence are presented for the schemes. Numerical experiments are performed to show the effectiveness of the schemes.

Positive Fitted Finite Volume Method for Semilinear Parabolic Systems on Unbounded Domain

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented.

Finite element method for a generalized constant delay diffusion equation

This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results.

Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes

It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As a consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic non-uniform mesh. We focus on grids mapped by a smooth function which consists in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix–vector product and detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter-free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power-graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability.

Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor’s series. Theoretical stability and convergence analysis is established in the L2-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.

Alternating direction implicit approach for the two-dimensional time fractional nonlinear Klein–Gordon and Sine–Gordon problems

The aim of this article is to establish a numerical scheme to achieve the theoretical accuracy near the weak singularity at t=0 in solving the two-dimensional time-fractional nonlinear mixed diffusion-wave equation (TFNMDWE). The governing problem involves diffusion term with time-fractional Caputo derivative (TFCD) of order ν1(0<ν1<1), and wave term with TFCD of order ν2(1<ν2<2). To handle the singularity at t=0, we use linearized L1 method to discretize both the TFCDs on nonuniform time meshes. By using the nonuniform L1 method to approximate TFCD and central difference operator for the space derivative approximation, the considered problem is converted to an equivalent system of equations. Then, we use the Alternating Direction Implicit (ADI) approach to develop a numerical scheme for solving the resulting system of coupled equations. Further, we prove stability analysis of the scheme. Numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFNMDWEs with smooth and non-smooth exact solutions to describe the accuracy of numerical scheme. The illustrated examples confirm that the scheme has second-order accuracy in space, and order of convergence (OC) in time direction is min(2−ν1,3−ν2,γν1,γ(ν2−1)), where γ is the mesh grading parameter used in construction of the nonuniform meshes. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equation

The aim of this work is to propose two efficient schemes to handle the accuracy near the singularity at t=0 in solving two-dimensional time-fractional diffusion-wave equation (TFDWE). The considered time fractional derivative is in the Caputo sense of order α(1<α<2). For the approximation of the time-fractional Caputo derivative (TFCD), we use Nonuniform L1 method (single-step) and Nonuniform Crank-Nicolson L1−2 method (multi-step). The L1 method has order of convergence (OC) min⁡(3−α,γα), where γ is the mesh grading parameter used in construction of the nonuniform mesh, and L1−2 method has second OC. We consider nonuniform time mesh to compensate the lack of smoothness caused by the presence of singularity in TFCD at t=0. After that, we adopt these two methods to approximate TFCD and apply the central difference operator for the space direction derivative approximations to get the system of equations for considered model. Then, we use the Alternating Direction Implicit (ADI) approach to develop two kinds of fully discrete schemes under the regularity conditions to solve the TFDWE. Further, we prove the stability analysis of these two schemes. Two numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFDWE with smooth and non-smooth exact solutions to indicate the accuracy of ADI schemes. The illustrated examples show that both schemes have second-order accuracy in space direction, and in temporal direction the schemes achieve min⁡(3−α,γα) and second order convergence, respectively for all 1<α<2. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

A new analytical approach for the local radial point interpolation discretisation in space and applications to high-order in time schemes for two-dimensional fractional PDEs

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity.

Hazard evaluation framework for large yield explosions in urban environments: A case study of Beirut explosion

The Beirut explosion is one of the largest yield ammonium nitrate explosions in recent years, causing massive destruction and becoming a valuable case for explosion analysis. Compared with general post-disaster evaluation methods, e.g., on-site investigation and high-resolution satellite imagery, the numerical simulation approach can provide decision-makers and rescue workers with intuitive and visualized assessment results. Additionally, it can also serve as an effective tool to guide the architectural planning and blast-resistant design of building structures. However, due to the large dimension of urban numerical models, simulating explosions in complex urban blocks has low efficiency, often costing days to ensure accuracy. At present, based on the finite element analysis software AUTODYN, a framework to evaluate the post-explosion damage in city blocks is proposed, which is composed of the geospatial data, efficient numerical simulation approach, and damage evaluation principles. Subsequently, the detailed process of the numerical simulation approach is given, and it integrates the multi-stage method, symmetrical modelling, optimized graded mesh size, mapping, and mesh un-refinement technique, to reduce the number of mesh cells and increase the computational efficiency. Furthermore, the applicability of the numerical simulation approach in more complex environments is verified through two typical block explosion tests. Finally, the Beirut explosion is reproduced using the geospatial data obtained from OpenStreetMap, and the evaluation results are compared with field survey reports, satellite image analysis, and related numerical simulations. It is found that the proposed framework can quickly and effectively predict and evaluate the actual explosion accidents in the city environment.

A general collocation analysis for weakly singular Volterra integral equations with variable exponent

Abstract Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-\alpha }$ for some constant $\alpha \in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng &amp; Wang (2020, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J. Numer. Anal., 58, 330–352), such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $\alpha = \alpha (t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper, the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed using novel techniques. These results then underpin an error analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel—it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.

A robust higher-order finite difference technique for a time-fractional singularly perturbed problem

A higher-order finite difference method is developed to solve the variable coefficients convection–diffusion singularly perturbed problems (SPPs) involving fractional-order time derivative with the order α∈(0,1). The solution to this problem class possesses a typical weak singularity at the initial time t=0 and an exponential boundary layer at the right lateral surface as the perturbation parameter ɛ→0. Alikhanov’s L2−1σ approximation is applied in the temporal direction on a suitable graded mesh, and the spatial variable is discretized on a piecewise uniform Shishkin mesh using a combination of midpoint upwind and central finite difference operators. Stability estimates and the convergence analysis of the fully discrete scheme are provided. It is shown that the fully discrete scheme is uniformly convergent with a rate of O(M−p+N−2(logN)2), where p=min{2,rα}, r is the graded mesh parameter and M,N are the number of mesh points in the time and space direction, respectively. Two numerical examples are taken in counter to confirm the sharpness of the theoretical estimates.

Error Estimates for Fractional Semilinear Optimal Control on Lipschitz Polytopes

We adopt the integral definition of the fractional Laplace operator and analyze solution techniques for fractional, semilinear, and elliptic optimal control problems posed on Lipschitz polytopes. We consider two strategies of discretization: a semidiscrete scheme where the admissible control set is not discretized and a fully discrete scheme where such a set is discretized with piecewise constant functions. As an instrumental step, we derive error estimates for finite element discretizations of fractional semilinear elliptic partial differential equations (PDEs) on quasi-uniform and graded meshes. With these estimates at hand, we derive error bounds for the semidiscrete scheme and improve the ones that are available in the literature for the fully discrete scheme.

A fully discrete scheme based on cubic splines and its analysis for time-fractional reaction–diffusion equations exhibiting weak initial singularity

The aim of this paper is to design and analyze a robust fully discrete scheme based on cubic splines for numerically solving a time-fractional reaction–diffusion equation (TFRDE) with smooth and non-smooth solutions. The solution of this problem exhibits a weak singularity near time t=0. The method combines the L1 scheme on a uniform or graded mesh to discretize in time and a cubic spline difference scheme on a uniform mesh to discretize in space. Further, the stability and convergence for both the smooth and non-smooth solutions are analyzed separately. Moreover, due to the use of a cubic spline difference scheme to discretize the space variable, the analysis of the fully discrete scheme requires some innovative ideas. In the end, two test problems are considered to demonstrate the validity and authenticity of the proposed method. The first example deals with the smooth solution, whereas in the second example, we consider a non-smooth solution.

Optimal error analysis of space–time second-order difference scheme for semi-linear non-local Sobolev-type equations with weakly singular kernel

In this paper, we construct and analyze a Crank–Nicolson difference scheme for solving the semilinear Sobolev-type equation with the Riemann–Liouville fractional integral of order α∈(0,1). The proposed scheme consists of two main stages. First, to compensate for the singular behavior of the exact solution at the initial time t=0, a Crank–Nicolson scheme and the product trapezoidal integration rule are proposed on graded meshes for time discretization. Second, a classical finite difference operator on uniform meshes is used for space discretization. Under suitable assumptions of the regularity condition, the stability and convergence analysis of this scheme in a new norm are given by the energy argument. It is shown that how the mesh grading exponent γ affects the temporal convergence rate of the presented scheme in the final convergence result. This scheme can attain a second-order accuracy in both time and space by choosing an optimal grading parameter γ. Numerical results confirm that the error analysis is sharp, and comparisons with results on uniform temporal meshes are included to indicate the effectiveness of our method.

Error analysis of a finite difference scheme on a modified graded mesh for a time-fractional diffusion equation

This paper is concerned with a finite difference scheme on a new modified graded mesh for a time fractional diffusion equation with a Caputo fractional derivative of order α∈(0,1). At first, the construction and some basic properties of this new modified graded mesh are investigated and on its basis the L1 scheme is applied to approximate the Caputo derivative. Meanwhile, the standard center finite difference scheme on a uniform mesh is used to discretize the diffusion term. Then, stability and convergence of the proposed scheme in the maximum norm are proved. The convergence result shows that on this modified graded mesh one attains an optimal 2−α rate for the L1 scheme. Finally, the presented theoretical results are supported by some numerical experiments.

Enhancing level set-based topology optimization with anisotropic graded meshes

We propose a new algorithm for the design of topologically optimized lightweight structures, under a minimum compliance requirement. The new process enhances a standard level set formulation in terms of computational efficiency, thanks to the employment of a computational mesh customized to the problem at hand. We pursue a twofold goal, i.e., to deliver a final layout characterized by a smooth contour and reliable mechanical properties. The smoothness of the optimized structure is ensured by the employment of an anisotropic adapted mesh, which sharply captures the material/void interface. A robust mechanical performance is guaranteed by a uniform tessellation of the internal part of the optimized configuration. A thorough numerical investigation corroborates the effectiveness of the proposed algorithm as a reliable and computationally affordable design tool, both in two- and three-dimensional contexts.