Corner Layers Research Articles

A parameter-uniform hybrid method for singularly perturbed parabolic 2D convection-diffusion-reaction problems

The solution of the singular perturbation problems (SPP) of convection-diffusion-reaction type may exhibit regular and corner layers in a rectangular domain. In this work, we construct and analyze a parameter-uniform operator-splitting alternating direction implicit (ADI) scheme to efficiently solve a two-dimensional parabolic singularly perturbed problem with two positive parameters. The proposed model is a combination of the backward-Euler method defined on a uniform mesh in time and a hybrid method in space defined on a special Shishkin mesh. The analysis is presented on a layer adapted piecewise-uniform Shishkin mesh. The developed numerical method is proved to be first-order convergent in time and almost second-order convergent in space. The numerical experiments are performed to validate the theoretical convergence results and illustrate the efficiency of the current strategy.

An integrated biorefinery strategy for Eucalyptus fractionation and co-producing glucose, furfural, and lignin based on deep eutectic solvent/cyclopentyl methyl ether system

A novel biphasic system containing water-soluble deep eutectic solvent (DES) and cyclopentyl methyl ether (CPME) was developed to treat Eucalyptus for furfural production, extracting lignin and enhancing cellulose enzymatic hydrolysis. Herein effect of DES type, water content in DES, temperature and time on furfural yield in water-soluble DES/CPME pretreatment process was firstly evaluated. A maximum furfural yield of 80.6 % was attained in 10 min at 150 °C with choline chloride (ChCl)/citric acid monohydrate (CAM)/CPME system containing 30 wt% water and 2.5 wt% SnCl4·5H2O, which was higher than that obtained from ChCl/CAM/CPME system without water (55.5 %) and H2O/CPME system (49.7 %). These results demonstrated that the water-soluble DES/CPME system was a powerful method enhancing the furfural production. Under the optimal pretreatment conditions, the delignification and glucose yield were reached to 72.7 % and 94.3 %, respectively. The extracted lignin showed low molecular weight and β-aryl-ether was obviously cleaved. Additionally, water-soluble DES/CPME pretreatment led to a significant removal of hemicelluloses (100.0 %) and lignin (72.7 %) and introduced morphological changes on cell walls, especially from the cell corner (CC) and secondary wall (SW) layers. Overall, this work proposed a practical one-step fractionation strategy for co-producing furfural, lignin and fermentable sugar, providing a way to biorefinery.

A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection–reaction–diffusion PDEs

In this work, we consider the numerical approximation of a two dimensional elliptic singularly perturbed weakly-coupled system of convection–reaction–diffusion type, which has two different parameters affecting the diffusion and the convection terms, respectively. The solution of such problems has, in general, exponential boundary layers as well as corner layers. To solve the continuous problem, we construct a numerical method which uses a finite difference scheme defined on an appropriate layer-adapted Bakhvalov–Shishkin mesh. Then, the numerical scheme is a first order uniformly convergent method with respect both convection and diffusion parameters. Numerical results obtained with the algorithm for some test problems are presented, which show the best performance of the proposed method, and they also corroborate in practice the theoretical analysis.

Numerical treatment of a singularly perturbed 2-D convection-diffusion elliptic problem with Robin-type boundary conditions

We consider a singularly perturbed two-dimensional steady-state convection-diffusion problem with Robin boundary conditions. The coefficient of the highest-order terms in the differential equation and in the boundary conditions, denoted by ϵ, is a positive perturbation parameter, and so it may be arbitrarily small. Solutions to such problems present regular (exponential) boundary layers as well as corner layers. In this article, a numerical approach is carried out using a finite-difference technique with an appropriate layer-adapted piecewise-uniform Shishkin mesh to provide a good approximation of the exact solution. Some numerical examples are presented that show that the approximations obtained are accurate and that they are in agreement with the theoretical results.

A finite difference method for a singularly perturbed 2‐D elliptic convection‐diffusion PDEs on Shishkin‐type meshes with non‐smooth convection and source terms

This paper deals with a class of singularly perturbed 2‐D elliptic convection‐diffusion PDEs with non‐smooth convection and source terms. The discontinuity in the convection and source terms portray corner and interior layers in the solution. This type of model problem often appears in modeling various physical phenomena, particularly, in mathematical biology, and thus requires effective numerical techniques for analyzing them computationally. For this purpose, we approximate the considered linear problem by developing an efficient numerical method. The spatial discretization for the numerical method is based on a finite difference scheme. The spatial domain is discretized by appropriate layer‐adapted meshes (S‐mesh and B‐S‐mesh) to accomplish this. The theoretical outcomes are finally supported by extensive numerical experiments, which also include a comparison of the proposed numerical method in terms of the order of accuracy and the computational cost.

A computational method for a two-parameter singularly perturbed elliptic problem with boundary and interior layers

In this article, we investigate a two-dimensional (2-D) singularly perturbed convection–reaction–diffusion elliptic type problem where two parameters ϵ and μ multiply the diffusion and convection terms, respectively. Furthermore, we assume that jump discontinuities exist in the source term along the x- and y-axis. Due to the presence of perturbation parameters, the solutions to such problems show boundary and corner layers. Moreover, the discontinuity in the source term adds the interior layers to the solution whose suitable numerical approach is the important goal of this article. A numerical approach is carried out using an upwind finite-difference technique that includes an appropriate layer-adapted piecewise uniform Shishkin mesh. Some examples are presented which show the good performance of the proposed method and the agreement with the theoretical analysis.

A higher-order finite difference method for two-dimensional singularly perturbed reaction-diffusion with source-term-discontinuous problem

This paper considers a two-dimensional singularly perturbed reaction-diffusion equation with a discontinuous source term. Due to this discontinuity, interior, corner, and boundary layers appear in the solution for adequately small values of the perturbation parameter ϵ. To achieve a decent estimate of the solution, we construct a numerical approach adopting an efficient hybrid finite difference method that includes a proper layer adapted piece-wise uniform Shishkin mesh. Further, we prove that the hybrid finite difference method is almost second-order uniformly convergent with respect to the perturbation parameter. We have implemented our method to test examples. Numerical results are verifying the theoretical results.

An analysis of diagonal and incomplete Cholesky preconditioners for singularly perturbed problems on layer-adapted meshes

We investigate the solution of linear systems of equations that arise when singularly perturbed reaction–diffusion partial differential equations are solved using a standard finite difference method on layer adapted grids. It is known that there are difficulties in solving such systems by direct methods when the perturbation parameter, $$\varepsilon $$ , is small (MacLachlan and Madden in SIAM J Sci Comput 35(5):A2225–A2254, 2013). Therefore, iterative methods are natural choices. However, we show that, in two dimensions, the condition number of the coefficient matrix grows unboundedly when $$\varepsilon $$ tends to zero, and so unpreconditioned iterative schemes, such as the conjugate gradient algorithm, perform poorly with respect to $$\varepsilon $$ . We provide a careful analysis of diagonal and incomplete Cholesky preconditioning methods, and show that the condition number of the preconditioned linear system is independent of the perturbation parameter. We demonstrate numerically the surprising fact that these schemes are more efficient when $$\varepsilon $$ is small, than when $$\varepsilon $$ is $$\mathcal {O}(1)$$ . Furthermore, our analysis shows that when the singularly perturbed problem features no corner layers, an incomplete Cholesky preconditioner performs extremely well when $$\varepsilon \ll 1$$ . We provide numerical evidence that our findings extend to three-dimensional problems.

Improved algorithm for calculating electric field in microwave applicators of irregular shapes with elements in translation

Our previous work shows that the use of two time-varying anisotropic layers on the basis of transformation optics can calculate the electric field in microwave applicators of regular shapes with moving elements in translation accurately and efficiently. However, it is difficult to deal with applicators of irregular shapes, which is widely used in practical applications. In this article, we extend this work to the calculation of electric field in applicators of irregular shapes. In the proposed two-dimensional model, the motion region is truncated by rectangular anisotropic coated layers, including four edge layers and four corner layers. In this case, the motion region is independent of applicator shapes. Therefore, the electric field can be calculated equivalently with arbitrary irregular applicator shapes. The results show good accuracy and efficiency, which indicates that the improved method has a good potential for modeling the heating process of practical microwave applicators with elements in translation.

Variational formulation, asymptotic analysis, and finite element simulation of wrinkling phenomena in modified plate equations modeling biofilms growing on agar substrates

The expansion of a bacterial biofilm on an agar substrate is modeled as a system of Föppl–von Kármán equations modified to include growth and coupling to a viscoelastic substrate. Analysis shows that wrinkles appear on the biofilm as a result of growth incompatibility and their frequency increases due to interaction with the agar layer. Simple cases of homogeneous radial and azimuthal growth are approximated by cone and corona solutions of the Monge–Ampère equation that are corrected by corner and boundary layers. A weak formulation of the problem allows us to express in-plane elastic strains and Airy potential solely in terms of the vertical displacement. We have developed a numerical method based on finite elements and simulated biofilm deformation for wide spectra of different growths. For heterogeneous growth, we find wrinkled patterns that are combinations of cones and coronas.

Robust exponential convergence of [formula omitted]-FEM in balanced norms for singularly perturbed reaction–diffusion problems: Corner domains

The hp-version of the finite element method is applied to singularly perturbed reaction–diffusion type equations on polygonal domains. The solution exhibits boundary layers as well as corner layers. On a class of meshes that are suitably refined near the boundary and the corners, robust exponential convergence (in the polynomial degree) is shown in both a balanced norm and the maximum norm.

Boundary layer analysis of nonlinear reaction–diffusion equations in a polygonal domain

We propose a boundary layer analysis which fits a domain with corners. In particular, we consider nonlinear reaction–diffusion problems posed in a polygonal domain having a small diffusive coefficient ε>0. We present the full analysis of the singular behaviours at any orders with respect to the parameter ε where we use a systematic nonlinear treatment initiated in Jung et al. (2016). The boundary layers are formed near the polygonal boundaries and two adjacent ones overlap at a corner P and the overlapping produces additional layers, the so-called corner layers. It is noteworthy that the boundary layers are also degenerate due to the singularities of the solutions involving a negative power of the radial distance to the corner P which are present in the Laplace operator on a sector (sector corresponding to the part of the polygon near the corner). The corner layers are then designed to absorb both the singularities and the interaction of the two boundary layers at P.

Boundary layers for the 3D primitive equations in a cube: The supercritical modes

In this article we study the boundary layers for the viscous Linearized Primitive Equations (LPEs) when the viscosity is small. The LPEs are considered here in a cube. Besides the usual boundary layers that we analyze here too, corner layers due to the interaction between the different boundary layers are also studied.

A multiscale sparse grid finite element method for a two-dimensional singularly perturbed reaction-diffusion problem

We consider the numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem posed on the unit square by a multiscale sparse grid finite element method. A Shishkin mesh which resolves the boundary and corner layers, and yields a parameter robust solution, is used. Our analysis shows that the method achieves essentially the same level of accuracy, in the energy norm, as the standard Galerkin finite element method with bilinear elements. However, only O(NlogN)$\mathcal {O}(N\log N)$ degrees of freedom are required, compared to O(N2)$\mathcal {O}(N^{2})$ for the corresponding Galerkin finite element method. This may be regarded as a generalisation of Liu et al. (IMA J. Numer. Anal. 29(4), 986---1007 2009) which used a two-scale method requiring O(N3/2)$\mathcal {O}(N^{3/2})$ degrees of freedom. Numerical results are provided that demonstrate the sharpness of the estimates and the efficiency of the method.

Influence of corner layers in the variational determination of bubble solutions of the constrained Allen–Cahn equation

A steady-state bubble solution to the constrained mass conserving Allen–Cahn equation in a two-dimensional domain is constructed in the limit of small diffusivity. The solution is asymptotically constant inside a circle of radius rb centred at some unknown location x0 and has a sharp interface at the bubble radius that allows for a transition to a different asymptotically constant state outside the bubble. In a study by M. J. Ward (Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math. 56, 1996, 247–1279), the bubble centre was determined by a limiting solvability condition. The solution found by Ward suggests the existence of a corner type boundary layer where the normal derivative of the bubble solution readjusts to satisfy the no-flux condition at the boundary of the domain. This work is concerned with the details of the readjustment. A variational approach similar to the one of W. L. Kath, C. Knessl and B. J. Matkowsky (A variational approach to nonlinear singularly perturbed boundary-value problems. Stud. Appl. Math. 77, 1987, 61–88) shows the formation of a corner layer (for the derivative of the solution) which influences as a high-order correction the available determination of the bubble centre. This corner layer describes to leading order the readjustment of the level lines of the bubble to lines parallel to the boundary of the container; moreover, it provides to leading order a smooth solution across the corner layer.

Finite elements scheme in enriched subspaces for singularly perturbed reaction–diffusion problems on a square domain

In this article, we discuss reaction-diffusion problems which produce ordinary boundary layers and elliptic corner layers. Using the classical polynomial Q1-finite elements spaces enriched with the so-called boundary layer elements which absorb the singularities due to the boundary and corner layers we are able to attain high numerical accuracies. We essentially obtain � -uniform approximation errors in a weighted energy norm with significant simplifications in the numerical implemen- tations; here we do not use mesh refinements.

An elliptic singularly perturbed problem with two parameters I: Solution decomposition

In this paper we consider a singularly perturbed elliptic problem with two small parameters posed on the unit square. Its solution may have exponential, parabolic and corner layers. We give a decomposition of the solution into regular and layer components and derive pointwise bounds on the components and their derivatives. The estimates are obtained by the analysis of appropriate problems on unbounded domains.

Numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem with discontinuous coefficients

We consider a singularly perturbed elliptic problem in two dimensions with discontinuous coefficients and line interface. The second-order derivatives are multiplied by a small parameter ε 2. The solution of this problem exhibits boundary, interior and corner layers. A finite volume difference scheme is constructed on partially uniform layer-adapted (Shishkin mesh). We prove that it yields an accurate approximation of the solution both inside and outside these layers. Error estimates in the discrete maximum norm that hold true uniformly in the perturbation parameter ε are obtained. Numerical experiments that agree with the theoretical result are given.

Non-modal stability and breakdown in corner and three-dimensional boundary layers

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Duck Peter W. and Owen Jonathan 2004Non-modal stability and breakdown in corner and three-dimensional boundary layersProc. R. Soc. Lond. A.4601335–1357http://doi.org/10.1098/rspa.2003.1215SectionRestricted accessNon-modal stability and breakdown in corner and three-dimensional boundary layers Peter W. Duck Peter W. Duck Department of Mathematics, University of Manchester, Manchester M13 9PL, UK () Google Scholar Find this author on PubMed Search for more papers by this author and Jonathan Owen Jonathan Owen Department of Mathematics, University of Manchester, Manchester M13 9PL, UK () Google Scholar Find this author on PubMed Search for more papers by this author Peter W. Duck Peter W. Duck Department of Mathematics, University of Manchester, Manchester M13 9PL, UK () Google Scholar Find this author on PubMed Search for more papers by this author and Jonathan Owen Jonathan Owen Department of Mathematics, University of Manchester, Manchester M13 9PL, UK () Google Scholar Find this author on PubMed Search for more papers by this author Published:08 May 2004https://doi.org/10.1098/rspa.2003.1215AbstractTwo classes of three-dimensional boundary layer are studied, namely, the flow in the neighbourhood of a corner (of general apex angle), and the flow over a wall with spanwise-varying porosity. Streamwise variation of the mainstream flow is assumed to be of the form xn, where x is a measure of the distance downstream of the leading edge, and as such the flow is analogous to the classical Falkner-Skan solution family. The main focus of the paper is on the effect of flow disturbances on the basic-flow states (with spanwise length-scales comparable with the boundary-layer thickness). The nature of the perturbation flow response is found to be inherently linked to the existence (or otherwise) of linear, steady, streamwise algebraically growing eigensolutions in the far-field solution. When nonlinearity is included, if such linear eigensolutions exist for the basic-flow state, downstream two scenarios appear possible: either the flow will make a transition to another base state (to one not possessing algebraically growing eigensolutions) or a breakdown may occur. Linear, unsteady (harmonic) perturbations are also considered. These are inherently non-parallel in nature (but again closely linked to the existence of the aforementioned eigensolutions); however, such disturbances always decay far downstream, although often not before a significant transient response has occurred. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Zhao W and Wang G (2017) Influence of acoustic resonance on mixing enhancement in confined mixing layers, Chemical Engineering and Processing: Process Intensification, 10.1016/j.cep.2016.11.005, 111, (67-78), Online publication date: 1-Jan-2017. Alizard F, Robinet J and Guiho F (2013) Transient growth in a right-angled streamwise corner, European Journal of Mechanics - B/Fluids, 10.1016/j.euromechflu.2012.07.006, 37, (99-111), Online publication date: 1-Jan-2013. Ridha A (2005) On the three-dimensional alternative to the Blasius boundary-layer solution, Comptes Rendus Mécanique, 10.1016/j.crme.2005.09.002, 333:10, (768-772), Online publication date: 1-Oct-2005. Duck P (2005) Transient growth in developing plane and Hagen Poiseuille flow, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461:2057, (1311-1333), Online publication date: 8-May-2005. This Issue08 May 2004Volume 460Issue 2045 Article InformationDOI:https://doi.org/10.1098/rspa.2003.1215Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/05/2004Published in print08/05/2004 License: Citations and impact Keywordscorner boundary layersbypass transitionthree-dimensional boundary layersnon-modal stability

Pattern formation in two-dimensional reactive-diffusive media with straining

The propagation of spiral waves in two-dimensional reactive-diffusive media subject to a velocity field with straining and no-flux boundary conditions is studied numerically. It is shown that, at low strain rates, the spiral wave preserves its integrity, whereas, at moderate ones, the spiral wave is deformed and transformed into a curved front which propagates almost along one of the principal directions of the strain rate tensor. At large strain rates, corner layers and elliptical fronts that become planar ones are observed.