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Martin Stynes - One of the best experts on this subject based on the ideXlab platform.
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regularity and derivative bounds for a convection Diffusion Problem with a neumann outflow condition
Journal of Differential Equations, 2009Co-Authors: Aidan Naughton, Bruce R Kellogg, Martin StynesAbstract:Abstract A convection–Diffusion Problem is considered on the unit square. The convective direction is parallel to two of the square's sides. A Neumann condition is imposed on the outflow boundary, with Dirichlet conditions on the other three sides. The precise relationship between the regularity of the solution and the global smoothness and corner compatibility of the data is elucidated. Pointwise bounds on derivatives of the solution are obtained; their dependence on the data regularity and compatibility and on the small Diffusion parameter is made explicit. The analysis uses Fourier transforms and Mikhlin multipliers to sharpen regularity results previously published for certain subProblems in a decomposition of the solution.
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a two scale sparse grid method for a singularly perturbed reaction Diffusion Problem in two dimensions
Ima Journal of Numerical Analysis, 2009Co-Authors: Fang Liu, Martin Stynes, Niall Madden, Aihui ZhouAbstract:The linear reaction-Diffusion Problem -e 2 Δu + bu = f is considered on the unit square with homogeneous Dirichlet boundary conditions. Here e is a small positive parameter and the Problem is in general singularly perturbed. The numerical solution of this Problem is analysed on a Shishkin mesh that has N intervals in each coordinate direction, using the Galerkin finite-element method with bilinear trial functions. The accuracy of this method, measured in the associated energy norm, is shown to be O(N -2 + e 1/2 N -1 lnN). It is proved that a two-scale sparse grid method achieves the same order of accuracy while reducing the number of degrees of freedom from O(N 2 ) to O(N 3/2 ). These results are then generalized to systems of reaction-Diffusion equations.
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sharpened bounds for corner singularities and boundary layers in a simple convection Diffusion Problem
Applied Mathematics Letters, 2007Co-Authors: Bruce R Kellogg, Martin StynesAbstract:Abstract A singularly perturbed convection–Diffusion Problem posed on the unit square is considered. Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present. Sharpened pointwise bounds on the solution and its derivatives are derived. The bounds improve bounds near an outflow corner of the Problem that were derived in an earlier paper of the authors. Application is made to an error analysis of a finite element method for the Problem.
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corner singularities and boundary layers in a simple convection Diffusion Problem
Journal of Differential Equations, 2005Co-Authors: Bruce R Kellogg, Martin StynesAbstract:Abstract A singularly perturbed convection–Diffusion Problem posed on the unit square is considered. Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present. Pointwise bounds on the solution and its derivatives are derived. The dependence of these bounds on the small Diffusion coefficient, on the regularity of the data, and on the compatibility of the data at the corners of the domain are all made explicit. The bounds are derived by decomposing the solution into a sum of solutions of elliptic boundary-value Problems posed on half-planes, then analyzing these simpler Problems.
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the sdfem for a convection Diffusion Problem with a boundary layer optimal error analysis and enhancement of accuracy
SIAM Journal on Numerical Analysis, 2003Co-Authors: Martin Stynes, Lutz TobiskaAbstract:The streamline-Diffusion finite element method (SDFEM) is applied to a convection-Diffusion Problem posed on the unit square, using a Shishkin rectangular mesh with piecewise bilinear trial functions. The hypotheses of the Problem exclude interior layers but allow exponential boundary layers. An error bound is proved for $\|u^I-u^N\|_{SD}$, where $u^I$ is the interpolant of the solution $u$, $u^N$ is the SDFEM solution, and $\|\cdot\|_{SD}$ is the streamline-Diffusion norm. This bound implies that $\|u-u^N\|_{L^2}$ is of optimal order, thereby settling an open question regarding the $L^2$-accuracy of the SDFEM on rectangular meshes. Furthermore, the bound shows that $u^N$ is superclose to $u^I$, which allows the construction of a simple postprocessing that yields a more accurate solution. Enhancement of the rate of convergence by using a discrete streamline-Diffusion norm is also discussed. Finally, the verification of these rates of convergence by numerical experiments is examined, and it is shown that this practice is less reliable than was previously believed.
Hao Cheng - One of the best experts on this subject based on the ideXlab platform.
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a regularization for a riesz feller space fractional backward Diffusion Problem
Inverse Problems in Science and Engineering, 2014Co-Authors: Hao Cheng, Chuli Fu, Guanghui ZhengAbstract:In the present paper, we consider a space-fractional backward Diffusion Problem, where data is given at and the solution is required in the interval . However, it is an ill-posed Problem since the solution does not depend continuously on the data. Here, we give a new iteration regularization method to deal with this Problem, and error estimates are obtained for a priori and a posteriori parameter choice rules, respectively. Furthermore, numerical implement shows the proposed method works effectively.
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an iteration regularization for a time fractional inverse Diffusion Problem
Applied Mathematical Modelling, 2012Co-Authors: Hao ChengAbstract:Abstract In the present paper we consider a time-fractional inverse Diffusion Problem, where data is given at x = 1 and the solution is required in the interval 0 x a priori and a posteriori parameter choice rules, respectively. Furthermore, numerical implement shows the proposed method works effectively.
Lutz Tobiska - One of the best experts on this subject based on the ideXlab platform.
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on the solution of the steady state Diffusion Problem for ferromagnetic particles in a magnetic fluid
Mathematical Modelling and Analysis, 2008Co-Authors: Viktor Polevikov, Lutz TobiskaAbstract:Abstract A mathematical model for the Diffusion process of ferromagnetic particles in a magnetic fluid is described. The unique solvability of the steady‐state particle concentration Problem is investigated and an analytical expression for its solution is found. In case that the fluid is under the action of a high‐gradient magnetic field a Stefan‐type Diffusion Problem can arise. An algorithm for solving the Stefan‐type steady‐state Problem is developed.
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the sdfem for a convection Diffusion Problem with a boundary layer optimal error analysis and enhancement of accuracy
SIAM Journal on Numerical Analysis, 2003Co-Authors: Martin Stynes, Lutz TobiskaAbstract:The streamline-Diffusion finite element method (SDFEM) is applied to a convection-Diffusion Problem posed on the unit square, using a Shishkin rectangular mesh with piecewise bilinear trial functions. The hypotheses of the Problem exclude interior layers but allow exponential boundary layers. An error bound is proved for $\|u^I-u^N\|_{SD}$, where $u^I$ is the interpolant of the solution $u$, $u^N$ is the SDFEM solution, and $\|\cdot\|_{SD}$ is the streamline-Diffusion norm. This bound implies that $\|u-u^N\|_{L^2}$ is of optimal order, thereby settling an open question regarding the $L^2$-accuracy of the SDFEM on rectangular meshes. Furthermore, the bound shows that $u^N$ is superclose to $u^I$, which allows the construction of a simple postprocessing that yields a more accurate solution. Enhancement of the rate of convergence by using a discrete streamline-Diffusion norm is also discussed. Finally, the verification of these rates of convergence by numerical experiments is examined, and it is shown that this practice is less reliable than was previously believed.
Eugene Oriordan - One of the best experts on this subject based on the ideXlab platform.
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a linearised singularly perturbed convection Diffusion Problem with an interior layer
Applied Numerical Mathematics, 2015Co-Authors: Eugene Oriordan, Jason Michael QuinnAbstract:Abstract A linear time dependent singularly perturbed convection–Diffusion Problem is examined. The convective coefficient contains an interior layer (with a hyperbolic tangent profile), which in turn induces an interior layer in the solution. A numerical method consisting of a monotone finite difference operator and a piecewise-uniform Shishkin mesh is constructed and analysed. Neglecting logarithmic factors, first order parameter uniform convergence is established.
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a parameter robust numerical method for a two dimensional reaction Diffusion Problem
Mathematics of Computation, 2005Co-Authors: C Clavero, J L Gracia, Eugene OriordanAbstract:In this paper a singularly perturbed reaction-Diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of Problems of this kind. A central finite difference scheme is constructed for this Problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.
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global maximum norm parameter uniform numerical method for a singularly perturbed convection Diffusion Problem with discontinuous convection coefficient
Mathematical and Computer Modelling, 2004Co-Authors: Paul A Farrell, Eugene Oriordan, Anthony F. Hegarty, John J. H. Miller, Grigorii I. ShishkinAbstract:A singularly perturbed convection-Diffusion Problem, with a discontinuous convection coefficient and a singular perturbation parameter @e, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this Problem, which generates @e-uniformly convergent numerical approximations to the solution. The method uses a piecewise uniform mesh, which is fitted to the interior layer, and the standard upwind finite difference operator on this mesh. The main theoretical result is the @e-uniform convergence in the global maximum norm of the approximations generated by this finite difference method. Numerical results are presented, which are in agreement with the theoretical results.
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a parameter uniform schwarz method for a singularly perturbed reaction Diffusion Problem with an interior layer
Applied Numerical Mathematics, 2000Co-Authors: John J. H. Miller, Eugene Oriordan, Grigorii I. Shishkin, Song WangAbstract:Abstract In this paper we consider numerical methods for a singularly perturbed reaction–Diffusion Problem with a discontinuous source term. We show that such a Problem arises naturally in the context of models of simple semiconductor devices. We construct a numerical method consisting of a standard finite difference operator and a non-standard piecewise-uniform mesh. The mesh is fitted to the boundary and interior layers that occur in the solution of the Problem. We show by extensive computations that, for this Problem, this method is parameter-uniform in the maximum norm, in the sense that the numerical solutions converge in the maximum norm uniformly with respect to the singular perturbation parameter.
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a uniformly convergent galerkin method on a shishkin mesh for a convection Diffusion Problem
Journal of Mathematical Analysis and Applications, 1997Co-Authors: Martin Stynes, Eugene OriordanAbstract:Abstract A Galerkin finite element method that uses piecewise bilinears on a simple piecewise equidistant mesh is applied to a linear convection-dominated convection-Diffusion Problem in two dimensions. The method is shown to be convergent, uniformly in the perturbation parameter, of orderN−1 ln Nin a global energy norm and of orderN−1/2ln3/2 Npointwise near the outflow boundary, where the total number of mesh points isO(N2).
Ercilia Sousa - One of the best experts on this subject based on the ideXlab platform.
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finite difference approximations for a fractional advection Diffusion Problem
Journal of Computational Physics, 2009Co-Authors: Ercilia SousaAbstract:The use of the conventional advection Diffusion equation in many physical situations has been questioned by many investigators in recent years and alternative Diffusion models have been proposed. Fractional space derivatives are used to model anomalous Diffusion or dispersion, where a particle plume spreads at a rate inconsistent with the classical Brownian motion model. When a fractional derivative replaces the second derivative in a Diffusion or dispersion model, it leads to enhanced Diffusion, also called superDiffusion. We consider a one-dimensional advection-Diffusion model, where the usual second-order derivative gives place to a fractional derivative of order @a, with 1<@a=<2. We derive explicit finite difference schemes which can be seen as generalizations of already existing schemes in the literature for the advection-Diffusion equation. We present the order of accuracy of the schemes and in order to show its convergence we prove they are stable under certain conditions. In the end we present a test Problem.