A Numerical Method for a Nonlinear Singularly Perturbed Interior Layer Problem Using an Approximate Layer Location

A class of nonlinear singularly perturbed interior layer problems is examined in this paper. Solutions exhibit an interior layer at an a priori unknown location. A numerical method is presented that uses a piecewise uniform mesh refined around approximations to the first two terms of the asymptotic expansion of the interior layer location. The first term in the expansion is used exactly in the construction of the approximation which restricts the range of problem data considered. The method is shown to converge point-wise to the true solution with a first order convergence rate (overlooking a logarithmic factor) for sufficiently small values of the perturbation parameter. A numerical experiment is presented to demonstrate the convergence rate established

Parameter-uniform numerical methods for singularly perturbed linear transport problems

Pointwise accurate numerical methods are constructed and analysed for three classes of singularly perturbed first order transport problems. The methods involve piecewise-uniform Shishkin meshes and the numerical approximations are shown to be parameter-uniformly convergent in the maximum norm. A transport problem from the modelling of fluid–particle interaction is formulated and used as a test problem for these numerical methods. Numerical results are presented to illustrate the performance of the numerical methods and to confirm the theoretical error bounds established in the paper. © 2022 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd

Convergence analysis for forward and inverse problems in singularly perturbed time-dependent reaction-advection-diffusion equations

In this paper, by employing the asymptotic expansion method, we prove the existence and uniqueness of a smoothing solution for a time-dependent nonlinear singularly perturbed partial differential equation (PDE) with a small-scale parameter. As a by-product, we obtain an approximate smooth solution, constructed from a sequence of reduced stationary PDEs with vanished high-order derivative terms. We prove that the accuracy of the constructed approximate solution can be in any order of this small-scale parameter in the whole domain, except a negligible transition layer. Furthermore, based on a simpler link equation between this approximate solution and the source function, we propose an efficient algorithm, called the asymptotic expansion regularization (AER), for solving nonlinear inverse source problems governed by the original PDE. The convergence-rate results of AER are proven, and the a posteriori error estimation of AER is also studied under some a priori assumptions of source functions. Various numerical examples are provided to demonstrate the efficiency of our new approach

On Approximating Discontinuous Solutions of PDEs by Adaptive Finite Elements

For singularly perturbed problems with a small diffusion, when the transient layer is very sharp and the computational mesh is relatively coarse, the solution can be viewed as discontinuous. For both linear and nonlinear hyperbolic partial differential equations, the solution can be discontinuous. When finite element methods with piecewise polynomials are used to approximate these discontinuous solutions, numerical solutions often overshoot near a discontinuity. Can this be resolved by adaptive mesh refinements? In this paper, for a simple discontinuous function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is the best choice to achieve accuracy without overshooting. For discontinuous piecewise linear approximations, non-trivial overshootings will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a "far away assumption", and non-trivial overshootings will always be observed under regular meshes. We calculate the explicit overshooting values for several typical cases. Several numerical tests are preformed for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper.Comment: 23 page

On The Two Dimensional Gierer-Meinhardt system with strong coupling

We construct solutions with a single interior condensation point for the two-dimensional Gierer-Meinhardt system with strong coupling. The condensation point is located at a nondegenerate critical point of the diagonal part of the regular part of the Green's function for -\Delta +1 nder the Neumann boundary condition. Our method is based on Liapunov-Schmidt reduction for a system of elliptic equations

Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case

Numerical computations often show that the Gierer-Meinhardt system has stable solutions which display patterns of multiple interior peaks (often also called spots). These patterns are also frequently observed in natural biological systems. It is assumed that the diffusion rate of the activator is very small and the diffusion rate of the inhibitor is finite (this is the so-called strong-coupling case). In this paper, we rigorously establish the existence and stability of such solutions of the full Gierer-Meinhardt system in two dimensions far from homogeneity. Green's function together with its derivatives plays a major role

An Upwind Finite Difference Method to Singularly Perturbed Convection Diffusion Problems on a Shishkin Mesh

This paper introduces a numerical approach to solve singularly perturbed convection diffusion boundary value problems for second-order ordinary differential equations that feature a small positive parameter {\epsilon} multiplying the highest derivative. We specifically examine Dirichlet boundary conditions. To solve this differential equation, we propose an upwind finite difference method and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. MATLAB code of the numerical recipe is made publicly available. We present numerical results to validate the theoretical results and assess the accuracy of our method. The tables and graphs included in this paper demonstrate the numerical outcomes, which indicate that our proposed method offers a highly accurate approximation of the exact solution.Comment: 19 pages, 4 figures. arXiv admin note: text overlap with arXiv:2305.18711 by other author

Interior layers in a reaction-diffusion equation with a discontinuous diffusion coefficient

In this paper a problem arising in the modelling of semiconductor devices motivates the study of singularly perturbed differential equations of reaction–diffusion type with discontinuous data. The solutions of such problems typically contain interior layers where the gradient of the solution changes rapidly. Parameter–uniform methods based on piecewise–uniform Shishkin meshes are constructed and analysed for such problems. Numerical results are presented to support the theoretical results and to illustrate the benefits of using a piecewise–uniform Shishkin mesh over the use of uniform meshes in the simulation of a simple semiconductor device