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Boundary and interior layer phenomena in coupled multiscale parabolic convection-diffusion interface problems: efficient numerical resolution and analysis

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INTERNATIONAL JOURNAL OF NUMERICAL METHODS FOR HEAT & FLUID FLOW 2025年

作者： Jaiswal, Aishwarya Kumar, Sunil Ramos, Higinio Indian Inst Technol BHU Varanasi Dept Math Sci Varanasi India Univ Salamanca Sci Comp Grp Salamanca Spain Univ Salamanca Escuela Politecn Super Zamora Salamanca Spain

PurposeThis paper aims to study boundary and interior layer phenomena in coupled multiscale parabolic convection-diffusion interface problems and to present their efficient numerical resolution and ***/methodology/approachThis study includes cases in which the diffusion parameters are small, distinct and can differ in order of magnitude. The source term is considered to be discontinuous. The asymptotic behavior of the solution is examined. The layer structure is analyzed, leading to the development of a variant of layer-resolving Shishkin mesh. For efficient numerical resolution, two methods are developed by combining additive schemes on a uniform mesh to discretize in time and an upwind difference scheme away from the line of discontinuity and a specific upwind difference scheme along the line of discontinuity, defined on a variant of layer resolving Shishkin mesh, to discretize in space. The analysis of the numerical resolution is discussed using the barrier function approach. Numerical simulations provide a verification of the theory and efficiency of the *** discontinuity in the source term, along with the inclusion of small and distinct diffusion parameters, results in multiple overlapping and interacting boundary and interior layers. The work demonstrates that the present approach is robust in resolving boundary and interior layers. From a computational cost perspective, the numerical resolution presented in the paper is more efficient than conventional ***/valueEfficient numerical resolution and analysis of boundary and interior layer phenomena in coupled multiscale parabolic convection-diffusion interface problems are provided. The discretization of the coupled system in the approach incorporates a distinctive feature, wherein the components of the approximate solution are decoupled at each time level, resulting in tridiagonal linear systems to be solved, in contrast to large banded linear systems with conventi

关键词： Boundary and interior layer phenomena Multiscale problems convection-diffusion problems Efficient numerical resolution Layer-adaptive meshes

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Superconvergence of discontinuous finite element solutions for transient convection-diffusion problems

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JOURNAL OF SCIENTIFIC COMPUTING 2005年第1期22-3卷 5-24页

作者： Adjerid, S Klauser, A Virginia Polytech Inst & State Univ Dept Math Blacksburg VA 24061 USA Tech Univ Munich D-85748 Munich Germany

We present a study of the local discontinuous Galerkin method for transient convection - diffusion problems in one dimension. We show that p-degree piecewise polynomial discontinuous finite element solutions of convection-dominated problems are O(Delta x(p+2)) superconvergent at Radau points. For diffusion-dominated problems, the solution's derivative is O(Delta x(p+2)) superconvergent at the roots of the derivative of Radau polynomial of degree p + 1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.

关键词： Discontinuous Galerkin method convection-diffusion problems superconvergence a posteriori error estimation

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A double boundary collocation Hermitian approach for the solution of steady state convection-diffusion problems

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2008年第9期55卷 1950-1960页

作者： La Rocca, A. Power, H. Univ Nottingham Sch Mech Mat & Manufacturing Engn Nottingham NG7 2RD England

In this article a double boundary collocation approach based on the meshless radial basis function Hermitian method (symmetric method) is proposed and compared with the conventional single collocation. In the double boundary collocation approach, at the boundary collocation points the boundary condition and the governing partial differential equation are required to be satisfied Simultaneously instead of only the boundary condition as required in the single collocation. We are able to carry out this type of algorithm due to the robustness of the proposed Hermite interpolation scheme, in which the resulting matrix will be non-singular as long as the partial differential operators applied to each point are linearly independent, even if in a single node we impose two different differential conditions. The results obtained with this new method are characterized by a higher precision especially for the prediction of the fluxes at the boundaries. This is due to the higher order of continuity of the approximation at the boundary points imposed by the double collocation. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： radial basis functions symmetric approach double collocation convection-diffusion problems

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Local projection methods on layer-adapted meshes for higher order discretisations of convection-diffusion problems

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APPLIED NUMERICAL MATHEMATICS 2009年第10期59卷 2515-2533页

作者： Matthies, Gunar Univ Kassel Fachbereich Math 17 D-34132 Kassel Germany

We consider singularly perturbed convection-diffusion problems in the unit square where the solutions show the typical exponential layers. Layer-adapted meshes (Shishkin and Bakhvalov-Shishkin meshes) and the local projection method are used to stabilise the discretised problem. Using enriched Q(r)-elements on the coarse part of the mesh and standard Q(r)-elements on the remaining parts of the mesh, we show that the difference between the solution of the stabilised discrete problem and a special interpolant of the solution of the continuous problem convergences F-uniformly with order O(N-(r+1/2)) on Bakhvalov-Shishkin meshes and with order O(N-(r+1/2) + N-(r+1) Inr+3/2 N) on Shishkin meshes. Furthermore, an epsilon-uniform convergence in the epsilon-weighted H-1-norm with order O((N-1 In N)(-r)) on Shishkin meshes and with order O(N-r) on Bakhvalov-Shishkin meshes will be proved. Numerical results which support the theory will be presented. (C) 2009 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： convection-diffusion problems Local projection method Layer-adapted mesh Quadrilateral finite elements

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MPI implementation of parallel subdomain methods for linear and nonlinear convection-diffusion problems

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JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING 2007年第5期67卷 581-591页

作者： Chau, Ming El Baz, Didier Guivarch, Ronan Spiteri, Pierre CNRS LAAS F-31077 Toulouse 4 France IRIT LIMA ENSEEIHT F-31071 Toulouse France

The solution of linear and nonlinear convection-diffusion problems via parallel subdomain methods is considered. MPI implementation of parallel Schwarz alternating methods on distributed memory multiprocessors is discussed. Parallel synchronous and asynchronous iterative schemes of computation are studied. Experimental results obtained from IBM-SP series machines are displayed and analyzed. The benefits of using parallel asynchronous Schwarz alternating methods are clearly shown. (C) 2007 Elsevier Inc. All rights reserved.

关键词： IBM-SP series machines MPI parallel computing asynchronous iterative algorithms Schwarz alternating method subdomain methods convection-diffusion problems

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Stable discretizations of convection-diffusion problems via computable negative-order inner products

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2000年第3期38卷 1034-1055页

作者： Bertoluzza, S Canuto, C Tabacco, A CNR Ist Anal Numer I-27100 Pavia Italy Politecn Torino Dipartimento Matemat I-10129 Turin Italy

A new functional framework for consistently stabilizing discrete approximations to convection-diffusion problems was recently proposed by the authors. The key ideas are the evaluation of the residual in an inner product of the type H-1/2 (unlike classical SUPG methods, which use elemental weighted L-2-inner products) and the realization of this inner product via explicitly computable multilevel decompositions of function spaces ( such as those given by wavelets or hierarchical finite elements). In this paper, we first provide further motivations for our approach. Next, we carry on a detailed analysis of the method, which covers all regimes (convection-dominated and diffusion-dominated). A consistent part of the analysis justifies the use of easily computable truncated forms of the stabilizing inner product. Numerical results, in close agreement with the theory, are given at the end of the paper.

关键词： singularly perturbed problems convection-diffusion problems stabilized Galerkin methods multiscale decompositions wavelets

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ON MIXED BOUNDARY ELEMENT SOLUTIONS OF convection-diffusion problems IN 3-DIMENSIONS

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APPLIED MATHEMATICAL MODELLING 1986年第3期10卷 170-175页

作者： TANAKA, Y HONMA, T KAJI, I Department of Electrical Engineering Faculty of Engineering Hokkaido University North 13 West 8 Sapporo 060Japan

Three-dimensional boundary element solutions of steady convection-diffusion problems are presented. Numerical properties such as stability and accuracy on boundary element solutions based on a discretization using mixed boundary elements are compared with those of constant elements. It is found that boundary element solutions are unconditionally stable in space, nd that their relative errors to exact solutions hardly depend on the Pecler number. These results prove that the boundary element method is superior to domain-type numerical techniques which have a criterion of numerical stability and whose approximate solutions depend to some extent on the Pecelet number. The advantage of BEM using mixed elements, as compared with constant element solutions is also shown.

关键词： mathematical model boundary element method convection-diffusion problems mixed element

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A fractional step ELLAM approach to high-dimensional convection-diffusion problems with forward particle tracking

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JOURNAL OF COMPUTATIONAL PHYSICS 2007年第1期221卷 198-225页

作者： Liang, Dong Du, Chuanbin Wang, Hong York Univ Dept Math & Stat Toronto ON M3J 1P3 Canada Univ S Carolina Dept Math Columbia SC 29208 USA

In this paper, a fractional step method combined with an Eulerian-Lagrangian localized adjoint method (ELLAM) is proposed to solve high-dimensional convection-diffusion problems. The method reduces high-dimensional problems to a series of uncoupled one-dimensional problems in each time step interval, in which one-dimensional ELLAM is used to solve the one-dimensional splitting equations. The approach takes the attractive advantages of the ELLAM method and the fractional step technique. It reduces computational complexities, large memory requirements, and long computation durations due to the application of the splitting technique. It reduces temporal errors and generates accurate numerical solutions even if large time and coarse spatial step sizes are used in computation. It effectively eliminates non-physical oscillation or excessive numerical dispersion and treats boundary conditions well and in a natural way. Numerical experiments show the efficient performance of the approach. (c) 2006 Elsevier Inc. All rights reserved.

关键词： convection-diffusion problems high-dimensions fractional step ELLAM

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A two-level variational multiscale meshless local Petrov-Galerkin (VMS-MLPG) method for convection-diffusion problems with large Peclet number

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COMPUTERS & FLUIDS 2018年 164卷 73-82页

作者： Chen, Zheng-Ji Li, Zeng-Yao Xie, Wen-Li Wu, Xue-Hong Xi An Jiao Tong Univ MOE Key Lab Thermal Fluid Sci & Engn Xian Shaanxi Peoples R China Zhengzhou Univ Light Ind Coll Electromech Sci & Engn Zhengzhou Henan Peoples R China

It is challengeable to obtain the stable and accurate solutions of convection-diffusion problems with large Peclet number (Pe) since the convection term may cause oscillation solutions at large Pe. In this paper, a unit operator (first level) and an orthogonal project operator (second level) are constructed to act as the stability terms for meshless local Petrov-Galerkin (MLPG) method, which is called a two-level variational multiscale MLPG (VMS-MLPG) method. The VMS-MLPG method is applied to eliminate oscillation, overshoots and undershoots of MLPG method at large Pe. The prediction accuracy and the numerical stability of the proposed method for the Smith-Hutton and the Brezzi problems are analyzed and validated by comparing with the MLPG method and the finite volume method (FVM) with various difference schemes. It is showed that the present VMS-MLPG method can guarantee the stable and reasonable solutions of convection-diffusion problems with large Peclet number. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： VMS-MLPG method convection-diffusion problems Large Peclet number

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UNIFORM SUPERCONVERGENCE OF A FINITE ELEMENT METHOD WITH EDGE STABILIZATION FOR convection-diffusion problems

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Journal of Computational Mathematics 2010年第1期28卷 32-44页

作者： Sebastian Franz Torsten Linβ Hans-Grg Roos Sebastian Schiller Department of Mathematics and Statistics University of Limerick Institut für Numerische Mathematik Technische Universitt Dresden

In the present paper the edge stabilization technique is applied to a convection-diffusion problem with exponential boundary layers on the unit square, using a Shishkin mesh with bilinear finite elements in the layer regions and linear elements on the coarse part of the mesh. An error bound is proved for ‖πu-u^h‖Е, where πu is some interpolant of the solution u and uh the discrete solution. This supercloseness result implies an optimal error estimate with respect to the L2 norm and opens the door to the application of postprocessing for improving the discrete solution.

关键词： convection-diffusion problems Edge stabilization FEM Uniform convergence Shishkin mesh.

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