The equation of interest in this work is the Richards equation which describes the flow of water in a porous medium. Since the equation contains two nonlinearities, stable and efficient methods have to be developed. I...
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A singularly perturbed parabolic equation of reaction-diffusion type is examined. Initially the solution approximates a concentrated source, which causes an interior layer to form within the solution for all future ti...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
A singularly perturbed parabolic equation of reaction-diffusion type is examined. Initially the solution approximates a concentrated source, which causes an interior layer to form within the solution for all future times. Combining a classical finite difference operator with a layer-adapted mesh, parameter-uniform convergence is established. Numerical results are presented to illustrate the theoretical error bounds.
The numerical solution of convection-diffusion-reaction equations in two and three dimensional domains Omega is thoroughly studied and well understood. Stabilized finite element methods have been developed to handle b...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
The numerical solution of convection-diffusion-reaction equations in two and three dimensional domains Omega is thoroughly studied and well understood. Stabilized finite element methods have been developed to handle boundary or interior layers and to localize and suppress unphysical oscillations. Much less is known about convection-diffusion-reaction equations on surfaces Gamma =partial derivative Omega. We propose a Local Projection Stabilization (LPS) for convection-diffusion-reaction equations on surfaces based on a linear surface approximation and first order finite elements. Unique solvability of the continuous and discrete problem are established. Numerical test examples show the potential of the proposed method.
The weak Galerkin (WG) methods are a class of nonconforming finite element methods, which were first introduced for a second order elliptic problem in Wang and Ye (2014). The idea of the WG is to introduce weak functi...
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We investigate the effect of interface irregularity on the convergence of the BDDC method for Navier-Stokes equations. A benchmark problem of a sequence of contracting channels is proposed to evaluate the robustness o...
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"Non-local" phenomena are common to problems involving strong heterogeneity, fracticality, or statistical correlations. A variety of temporal and/or spatial fractional partial differential equations have bee...
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ISBN:
(数字)9783319519548
ISBN:
(纸本)9783319519548;9783319519531
"Non-local" phenomena are common to problems involving strong heterogeneity, fracticality, or statistical correlations. A variety of temporal and/or spatial fractional partial differential equations have been used in the last two decades to describe different problems such as turbulent flow, contaminant transport in ground water, solute transport in porous media, and viscoelasticity in polymer materials. The study presented herein is focused on the numerical solution of spatial fractional advection-diffusion equations (FADEs) via the reproducing kernel particle method (RKPM), providing a framework for the numerical discretization of spacial FADEs. However, our investigation found that an alternative formula of the Caputo fractional derivative should be used when adopting Gauss quadrature to integrate equations with fractional derivatives. Several one-dimensional examples were devised to demonstrate the effectiveness and accuracy of the RKPM and the alternative formula.
Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H1 seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss anisotropic problems, semilinear equations, supercloseness and a combination technique. Moreover, we consider different classes of layer-adapted meshes and sketch the three-dimensional case. Remarks to systems and problems with different layers close the paper.
Residual-type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
Residual-type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction-diffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. The error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter. The case of the Dirichlet boundary conditions was considered in the recent article (Kopteva, Numer. Math., 2017, Published online 2 May 2017. doi:10.1007/s00211-017-0889-3). Now we extend this analysis to also allow boundary conditions of Neumann type.
We propose nonoverlapping domain decomposition methods for solving the total variation minimization problem. We decompose the domain of the dual problem into nonoverlapping rectangular subdomains, where local total va...
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Recently, several classes of fourth order singularly perturbed problems were considered and uniform convergence in the associated energy norm as well as in a balanced norm was proved. In this proceedings paper we will...
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ISBN:
(数字)9783319672021
ISBN:
(纸本)9783319672021;9783319672014
Recently, several classes of fourth order singularly perturbed problems were considered and uniform convergence in the associated energy norm as well as in a balanced norm was proved. In this proceedings paper we will extend some results by looking into L-infinity-bounds and postprocessing.
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