1,721,000 research outputs found
Enhanced RFB method
No full textThe residual-free bubble method (RFB) is a parameter-free stable finite element method that has been successfully applied to a wide range of boundary-value problems exhibiting multiple-scale behaviour. If some local features of the solution are known a priori, the approximation properties of the RFB finite element space can be improved by enriching it on selected edges of the partition by edge-bubbles that are supported on pairs of neighbouring elements. Motivated by this idea, we define and analyse an enhanced residual-free bubble method for the solution of convection-dominated convection-diffusion problems in 2-D. Our a priori analysis highlights the limitations of the RFB method and the improved global approximation properties of the new method. The theoretical results are supported by detailed numerical experiments. © Springer-Verlag 2005
The Residual-Free-Bubble Finite Element Method on Anisotropic Partitions.
Full text linkThe subject of this work is the analysis and implementation of stabilized finite element methods on anisotropic meshes. We develop the anisotropic a priori error analysis of the residual-free-bubble (RFB) method applied to elliptic convection-dominated convection-diffusion problems in two dimensions, with finite element spaces of type Qk., k ≥ 1. In the case of P1 finite elements, relying on the equivalence of the RFB method to classical stabilized finite element methods, we propose a new rule, justified through the analysis of the RFB method, for selecting the stabilization parameter in classical stabilized methods on two-dimensional anisotropic triangulations. © 2007 Society for Industrial and Applied Mathematics
Flux reconstruction and solution post-processing in mimetic finite difference methods
No full textWe present a post-processing technique for the mimetic finite difference solution of diffusion problems in mixed form. Our post-processing method yields a piecewise linear approximation of the scalar variable that is second-order accurate in the L2-norm on quite general polyhedral meshes, including non-convex and non-matching elements. The post-processing is based on the reconstruction of vector fields projected onto the mimetic space of vector variables. This technique is exact on constant vector fields and is shown to be independent of the mimetic scalar product choice if a local consistency condition is satisfied. The post-processing method is computationally inexpensive. Optimal performance is confirmed by numerical experiments. © 2007 Elsevier B.V. All rights reserved
Enhanced residual-free bubble method for convection-diffusion problems
No full textWe analyse the performance of the enhanced residual-free bubble (RFBe) method for the solution of elliptic convection-dominated convection-diffusion problems in 2-D, and compare the present method with the standard residual-free bubble (RFB) method. The advantages of the RFBe method are two-fold: it has better stability properties and it can be used to resolve boundary layers with high accuracy on globally coarse meshes. Copyright © 2005 John Wiley and Sons, Ltd
Hp-Version discontinuous galerkin methods on essentially arbitrarily-shaped elements
No full textWe extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for emph{curved} element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace and Markov-type inverse estimates to arbitrary element shapes. A further new H1−L2-type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach
Convergence analysis of the mimetic finite difference method for elliptic problems
No full textWe propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms. Our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes. The a priori error analysis relies on the connection between the mimetic formulation and the lowest order Raviart-Thomas mixed finite element method. The theoretical results are confirmed by numerical experiments. © 2009 Society for Industrial and Applied Mathematics
A Posteriori Error Analysis for Implicit–Explicit hp-Discontinuous Galerkin Timestepping Methods for Semilinear Parabolic Problems
Full text linkA posteriori error estimates in the L∞(H) - and L2(V) -norms are derived for fully-discrete space–time methods discretising semilinear parabolic problems; here V↪ H↪ V∗ denotes a Gelfand triple for an evolution partial differential equation problem. In particular, an implicit–explicit variable order (hp-version) discontinuous Galerkin timestepping scheme is employed, in conjunction with conforming finite element discretisation in space. The nonlinear reaction is treated explicitly, while the linear spatial operator is treated implicitly, allowing for time-marching without the need to solve a nonlinear system per timestep. The main tool in obtaining these error estimates is a recent space–time reconstruction proposed in Georgoulis et al. (A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems, Submitted for publication) for linear parabolic problems, which is now extended to semilinear problems via a non-standard continuation argument. Some numerical investigations are also included highlighting the optimality of the proposed a posteriori bounds
Hp-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes
No full textWe present a new hp-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of total degree, say p, defined in the physical coordinate system, as opposed to standard dG time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete hp-dG scheme using fewer degrees of freedom for each time step, compared to dG time-stepping schemes employing a tensorized space-time basis, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with an arbitrary number of faces. A priori error bounds are shown for the proposed method in various norms. An extensive comparison among the new space-time dG method, the (standard) tensorized space-time dG methods, the classical dG time-stepping, and the conforming finite element method in space is presented in a series of numerical experiments
A fast algorithm for determining the propagation path of multiple diffracted rays
No full textWe present a fast algorithm for path computation of multiple diffracted rays relevant to ray tracing techniques. The focus is on double diffracted rays, but generalizations are also mentioned. The novelty of our approach is in the use of an analytical geometry procedure which permits to re-write the problem as a simple nonlinear equation. This procedure permits a convergence analysis of the algorithms involved in the numerical resolution of such nonlinear equation. Moreover, we also indicate how to choose the iteration starting point to obtain convergence of the (locally convergent) Newton method. As in previous works, explicit solutions are obtained in the relevant cases of parallel or incident diffraction edges. © 2007 IEEE
A comparison of non-matching techniques for the finite element approximation of interface problems
No full textWe perform a systematic comparison of various numerical schemes for the approximation of interface problems. We consider unfitted approaches in view of their application to possibly moving configurations. Particular attention is paid to the implementation aspects and to the analysis of the costs related to the different phases of the simulations
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