BAIL 2006 Book of Abstracts - Institut für Numerische und ...
BAIL 2006 Book of Abstracts - Institut für Numerische und ... BAIL 2006 Book of Abstracts - Institut für Numerische und ...
A.CANGIANI, E.H.GEORGOULIS, M. JENSEN: Continuous-Discontinuous Finite Element Methods for Convection-Diffusion Problems ✬ ✫ References [1] Brezzi, F., Marini, D., and Süli, E. Residual-free bubbles for advection-diffusion problems: the general error analysis. Numer. Math. 85, 1 (2000), 31–47. [2] Brezzi, F., and Russo, A. Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4, 4 (1994), 571–587. [3] Brooks, A. N., and Hughes, T. J. R. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 32, 1-3 (1982), 199–259. FENOMECH ’81, Part I (Stuttgart, 1981). [4] Cangiani, A., and Süli, E. Enhanced RFB method. Numer. Math. 101(2) (2005), 273–308. [5] Eriksson, K., Estep, D., Hansbo, P., and Johnson, C. Introduction to adaptive methods for differential equations. In Acta numerica, 1995, Acta Numer. Cambridge Univ. Press, Cambridge, 1995, pp. 105–158. [6] Giles, M. B., and Süli, E. Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. In Acta numerica, 2002, vol. 11 of Acta Numer. 2002, pp. 145–236. [7] Hughes, Thomas J. R.and Scovazzi, G., Bochev, P. B., and Buffa, A. A multiscale discontinuous Galerkin method with the computational structure of a continuous galerkin method. ICES Report 05-16 (2005). [8] Hughes, T. J. R., and Brooks, A. A multidimensional upwind scheme with no crosswind diffusion. In Finite element methods for convection dominated flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979), vol. 34 of AMD. Amer. Soc. Mech. Engrs. (ASME), New York, 1979, pp. 19–35. [9] Hughes, T. J. R., Feijóo, G. R., Mazzei, L., and Quincy, J.-B. The variational multiscale method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 166, 1-2 (1998), 3–24. [10] Hughes, T. J. R., and Stewart, J. R. A space-time formulation for multiscale phenomena. J. Comput. Appl. Math. 74, 1-2 (1996), 217–229. [11] Madden, N., and Stynes, M. Efficient generation of shishkin meshes in solving convectiondiffusion problems. Preprint of the Department of Mathematics, University College, Cork, Ireland no. 1995-2 (1995). [12] Morton, K. W. Numerical solution of convection-diffusion problems, vol. 12 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London, 1996. [13] O’Riordan, E., and Stynes, M. A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions. Math. Comp. 57, 195 (1991), 47–62. [14] Roos, H.-G., Stynes, M., and Tobiska, L. Numerical Methods for Singularly Perturbed Differential Equations. Springer-Verlag, Berlin, 1996. Convection-diffusion and flow problems. [15] Schwab, C., and Suri, M. The p and hp versions of the finite element method for problems with boundary layers. Math. Comp. 65, 216 (1996), 1403–1429. 2 Speaker: GEORGOULIS, E.H. 76 BAIL 2006 ✩ ✪
C. CLAVERO, J.L. GRACIA, F. LISBONA: A second order uniform convergent method for a singularly perturbed parabolic system of reaction-diffusion type ✬ ✫ A second order uniform convergent method for a singularly perturbed parabolic system of reaction–diffusion type ∗ C. Clavero, J.L. Gracia and F. Lisbona Department of Applied Mathematics University of Zaragoza. Spain clavero@unizar.es, jlgracia@unizar.es, lisbona@unizar.es Abstract In this work we are interested in solving singularly perturbed parabolic boundary value problems given by ⎧ ⎪⎨ L�ε�u ≡ ⎪⎩ ∂�u ∂t + Lx,�ε�u = � f, (x, t) ∈ Q = Ω × (0, T ] = (0, 1) × (0, T ], �u(0, t) = �g0(t), �u(1, t) = �g1(t), ∀t ∈ [0, T ], (1) �u(x, 0) = �0, ∀x ∈ Ω, where Lx,�ε ≡ � −ε1 ∂2 ∂x 2 −ε2 ∂2 ∂x 2 � � � a11(x) a12(x) + A, A = , a21(x) a22(x) with 0 < ε1 ≤ ε2 ≤ 1. The components of the functions �g0(t), �g1(t), the right hand side �f(x, t) = (f1(x, t), f2(x, t)) T and the matrix A are supposed smooth enough. The positivity condition ai1 + ai2 ≥ αi > 0, aii > 0, i = 1, 2, aij ≤ 0 if i �= j, (2) is satisfied by matrix A. Otherwise, we consider the transformation �v(x, t) = �u(x, t)e −α0t with α0 > 0 sufficiently large so that (2) holds. This problem is a simple model of the classical linear double–diffusion model for saturated flow in fractures porous media (Barenblatt system) developed in [1]. The first equation describes the flow in the matrix and the second one the flow in the fracture system. The coupling terms are given by the matrix � � 1 −1 A = , −1 1 which describes the interchange of fluid between the two systems. The permeabilities ε1 and ε2 in these equations could be very small and also they can have different magnitudes. It is well known that the exact solution of these problems has a multiscale character, i.e., there are narrow regions (the boundary layer regions) where the solution has strong gradients and in the rest of the domain the solution varies smoothly. Therefore, it is necessary to dispose of efficient numerical methods (uniformly convergent methods) to approximate the solution independently of the values of the diffusion parameters ε1 and ε2. Recently some papers study uniform convergent numerical methods to solve singularly perturbed elliptic linear systems on a special piecewise uniform Shishkin mesh. In the analysis of these type of problems the three following different cases can be distinguished depending on the relation between the singular perturbation parameters ε1 and ε2: [i.] ε1 = ε, ε2 = 1, [ii.] ε1 = ε2 = ε [iii.] ε1, ε2 arbitrary. ∗ This research will be partially supported by the project MEC/FEDER MTM 2004-01905 and the Diputación General de Aragón. 1 Speaker: CLAVERO, C. 77 BAIL 2006 ✩ ✪
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A.CANGIANI, E.H.GEORGOULIS, M. JENSEN: Continuous-Discontinuous Finite<br />
Element Methods for Convection-Diffusion Problems<br />
✬<br />
✫<br />
References<br />
[1] Brezzi, F., Marini, D., and Süli, E. Residual-free bubbles for advection-diffusion problems:<br />
the general error analysis. Numer. Math. 85, 1 (2000), 31–47.<br />
[2] Brezzi, F., and Russo, A. Choosing bubbles for advection-diffusion problems. Math. Models<br />
Methods Appl. Sci. 4, 4 (1994), 571–587.<br />
[3] Brooks, A. N., and Hughes, T. J. R. Streamline upwind/Petrov-Galerkin formulations<br />
for convection dominated flows with particular emphasis on the incompressible Navier-Stokes<br />
equations. Comput. Methods Appl. Mech. Engrg. 32, 1-3 (1982), 199–259. FENOMECH ’81,<br />
Part I (Stuttgart, 1981).<br />
[4] Cangiani, A., and Süli, E. Enhanced RFB method. Numer. Math. 101(2) (2005), 273–308.<br />
[5] Eriksson, K., Estep, D., Hansbo, P., and Johnson, C. Introduction to adaptive methods<br />
for differential equations. In Acta numerica, 1995, Acta Numer. Cambridge Univ. Press,<br />
Cambridge, 1995, pp. 105–158.<br />
[6] Giles, M. B., and Süli, E. Adjoint methods for PDEs: a posteriori error analysis and<br />
postprocessing by duality. In Acta numerica, 2002, vol. 11 <strong>of</strong> Acta Numer. 2002, pp. 145–236.<br />
[7] Hughes, Thomas J. R.and Scovazzi, G., Bochev, P. B., and Buffa, A. A multiscale<br />
discontinuous Galerkin method with the computational structure <strong>of</strong> a continuous galerkin<br />
method. ICES Report 05-16 (2005).<br />
[8] Hughes, T. J. R., and Brooks, A. A multidimensional upwind scheme with no crosswind<br />
diffusion. In Finite element methods for convection dominated flows (Papers, Winter Ann.<br />
Meeting Amer. Soc. Mech. Engrs., New York, 1979), vol. 34 <strong>of</strong> AMD. Amer. Soc. Mech. Engrs.<br />
(ASME), New York, 1979, pp. 19–35.<br />
[9] Hughes, T. J. R., Feijóo, G. R., Mazzei, L., and Quincy, J.-B. The variational multiscale<br />
method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg.<br />
166, 1-2 (1998), 3–24.<br />
[10] Hughes, T. J. R., and Stewart, J. R. A space-time formulation for multiscale phenomena.<br />
J. Comput. Appl. Math. 74, 1-2 (1996), 217–229.<br />
[11] Madden, N., and Stynes, M. Efficient generation <strong>of</strong> shishkin meshes in solving convectiondiffusion<br />
problems. Preprint <strong>of</strong> the Department <strong>of</strong> Mathematics, University College, Cork,<br />
Ireland no. 1995-2 (1995).<br />
[12] Morton, K. W. Numerical solution <strong>of</strong> convection-diffusion problems, vol. 12 <strong>of</strong> Applied Mathematics<br />
and Mathematical Computation. Chapman & Hall, London, 1996.<br />
[13] O’Riordan, E., and Stynes, M. A globally uniformly convergent finite element method for<br />
a singularly perturbed elliptic problem in two dimensions. Math. Comp. 57, 195 (1991), 47–62.<br />
[14] Roos, H.-G., Stynes, M., and Tobiska, L. Numerical Methods for Singularly Perturbed<br />
Differential Equations. Springer-Verlag, Berlin, 1996. Convection-diffusion and flow problems.<br />
[15] Schwab, C., and Suri, M. The p and hp versions <strong>of</strong> the finite element method for problems<br />
with bo<strong>und</strong>ary layers. Math. Comp. 65, 216 (1996), 1403–1429.<br />
2<br />
Speaker: GEORGOULIS, E.H. 76 <strong>BAIL</strong> <strong>2006</strong><br />
✩<br />
✪