PDF superior TítuloA posteriori error analysis of an augmented mixed finite element method for Darcy flow

TítuloA posteriori error analysis of an augmented mixed finite element method for Darcy flow

TítuloA posteriori error analysis of an augmented mixed finite element method for Darcy flow

The outline of the paper is as follows. In Section 2 we first describe the problem of Darcy’s flow and recall its classical dual-mixed variational formulation. Then, we introduce a slight generaliza- tion of the second method analyzed in [15] and provide sufficient conditions on the stabilization parameters that allow to guarantee that the augmented variational formulation is well-posed. In Section 3 we describe the augmented discrete scheme and analyze its stability and convergence properties. We also provide the corresponding rate of convergence for some specific finite element subspaces. The new a posteriori error estimator is derived in Section 4, where we also prove that it is reliable and locally efficient. Finally, some numerical experiments are reported in Section 5. Conclusions are drawn in Section 6.
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20 Lee mas

TítuloA posteriori error analysis of an augmented mixed formulation in linear elasticity with mixed and Dirichlet boundary conditions

TítuloA posteriori error analysis of an augmented mixed formulation in linear elasticity with mixed and Dirichlet boundary conditions

Although the usual dual-mixed variational formulations satisfy the hypotheses of the Babuˇska-Brezzi theory, it is difficult to derive explicit finite element subspaces yield- ing stable discrete schemes. In particular, when mixed boundary conditions with non- homogeneous Neumann data are imposed, the PEERS elements can be applied but they yield a non-conforming Galerkin scheme. This was one of the main motivations to intro- duce the augmented formulation from [17].

31 Lee mas

Primal and mixed finite element methods for image registration

Primal and mixed finite element methods for image registration

Deformable Image Registration (DIR) is a powerful computational method for image analysis, with promising applications in the diagnosis of human disease. Despite being widely used in the medical imaging community, the mathematical and numerical analysis of DIR methods still has many open questions. Further, recent applications of DIR in- clude the quantification of mechanical quantities in addition to the aligning transformation, which justifies the development of novel DIR formulations for which the accuracy and convergence of fields other than the aligning transformation can be studied. In this work we propose and analyze a primal, mixed and augmented formulations for the DIR prob- lem, together with their finite-element discretization schemes for their numerical solution. The DIR variational problem is equivalent to the linear elasticity problem with a source term that has a nonlinear dependence on the unknown field. Fixed point arguments and small data assumptions are employed to derive the well-posedness of both the continuous and discrete schemes for the usual primal and mixed variational formulations, as well as for an augmented version of the later. In particular, continuous piecewise linear elements for the displacement in the case of the primal method, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) for the stress together with piecewise constants (resp. continuous piecewise linear) for the displacement when using the mixed approach (resp. its augmented version), constitute feasible choices that guarantee the stability of the asso- ciated Galerkin systems. A priori error estimates derived by using Strang-type lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of the method.
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71 Lee mas

TítuloLow cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

TítuloLow cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori er- ror estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illus- trate the performance of the corresponding adaptive algorithms and support the theoretical results.
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38 Lee mas

TítuloA low order mixed finite element method for a class of quasi Newtonian Stokes flows  Part I: a priori error analysis

TítuloA low order mixed finite element method for a class of quasi Newtonian Stokes flows Part I: a priori error analysis

In the recent papers [3,16] we analyzed dual-mixed formulations for non-linear boundary value problems in plane elasticity. In the case of incompressible materials, we considered the non-Newtonian model from [5,7], and applied the dual-mixed approach from [11] to study its solvability and finite element approxi- mations. Since the non-linear constitutive law depends on the strain tensor, we introduced this variable and the rotation as further unknowns, which yielded a twofold saddle point operator equation as the resulting variational formulation. Then, we extended the well known PEERS space and defined a stable Galerkin scheme, for which a Bank–Weiser type a posteriori error analysis was also developed.
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12 Lee mas

Primal and Mixed Finite Element Methods for Deformable Image Registration Problems

Primal and Mixed Finite Element Methods for Deformable Image Registration Problems

Deformable image registration (DIR) represent a powerful computational method for image analy- sis, with promising applications in the diagnosis of human disease. Despite being widely used in the medical imaging community, the mathematical and numerical analysis of DIR methods remain un- derstudied. Further, recent applications of DIR include the quantification of mechanical quantities apart from the aligning transformation, which justifies the development of novel DIR formulations where the accuracy and convergence of fields other than the aligning transformation can be stud- ied. In this work we propose and analyze a primal, mixed and augmented formulations for the DIR problem, together with their finite-element discretization schemes for their numerical solution. The DIR variational problem is equivalent to the linear elasticity problem with a nonlinear source term that depends on the unknown field. Fixed point arguments and small data assumptions are em- ployed to derive the well-posedness of both the continuous and discrete schemes for the usual primal and mixed variational formulations, as well as for an augmented version of the later. In particular, continuous piecewise linear elements for the displacement in the case of the primal method, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) for the stress together with piecewise constants (resp. continuous piecewise linear) for the displacement when using the mixed approach (resp. its augmented version), constitute feasible choices that guarantee the stability of the associated Galerkin systems. A-priori error estimates derived by using Strang-type Lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of the method.
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27 Lee mas

TítuloA mixed finite element method for the generalized Stokes problem

TítuloA mixed finite element method for the generalized Stokes problem

We present and analyse a new mixed finite element method for the generalized Stokes prob- lem. The approach, which is a natural extension of a previous procedure applied to quasi- Newtonian Stokes flows, is based on the introduction of the flux and the tensor gradient of the velocity as further unknowns. This yields a two-fold saddle point operator equation as the resulting variational formulation. Then, applying a slight generalization of the well known Babuˇ ska-Brezzi theory, we prove that the continuous and discrete formulations are well posed, and derive the associated a priori error analysis. In particular, the finite element subspaces providing stability coincide with those employed for the usual Stokes flows except for one of them that needs to be suitably enriched. We also develop an a-posteriori error estimate (based on local problems) and propose the associated adaptive algorithm to com- pute the finite element solutions. Several numerical results illustrate the performance of the method and its capability to localize boundary layers, inner layers, and singularities.
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27 Lee mas

TítuloA low order mixed finite element method for a class of quasi Newtonian Stokes flows  Part II: a posteriori error analysis

TítuloA low order mixed finite element method for a class of quasi Newtonian Stokes flows Part II: a posteriori error analysis

On the other hand, we recall that the application of adaptive algorithms, based on a posteriori error estimates, usually guarantees the quasi-optimal rate of convergence of the finite element solution to boundary value problems. In addition, this adaptivity is specially necessary for nonlinear problems where no a priori hints on how to build suitable meshes are available. To this respect, we have shown recently that the combination of the usual Bank–Weiser approach from [1] with the analysis from [3,4] allows to derive fully explicit and reliable a posteriori error estimates for the dual-mixed variational formulations (showing a two- fold saddle point structure) of some linear and nonlinear problems (see, e.g. [2,6,7]). However, no a pos- teriori error analysis has been developed yet for the nonlinear Stokes problems studied in [5]. Therefore, as a natural continuation of our results in [5], in the present paper we apply the Bank–Weiser type a posteriori error analysis mentioned above to derive reliable estimates for the mixed finite element scheme (1.6). The rest of this work is organized as follows. In Section 2 we collect some basic results on Sobolev spaces and state the main result of this paper. The proof of our a posteriori estimate, which makes use of the Ritz projection of the error, is provided in Section 3. In Section 4 we prove the quasi-efficiency of the estimator and discuss on suitable choices for the auxiliary functions needed for its computation. Finally, several numerical results illustrating the good performance of the adaptive algorithm are reported in Section 5.
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19 Lee mas

TítuloA residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity

TítuloA residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity

According to the above, and strongly motivated by the competitive character of our augmented formulation, we now feel the need of deriving corresponding a posteriori error estimators. More precisely, the purpose of this work is to develop a residual based a posteriori error analysis for the augmented mixed finite element scheme from [10] in the case of pure Dirichlet boundary conditions. A posteriori error analyses of the traditional mixed finite element methods for the elasticity problem can be seen in [5] and the references therein. The rest of this paper is organized as follows. In Section 2 we recall from [10] the continuous and discrete augmented formulations of the corresponding boundary value problem, state the well-posedness of both schemes, and provide the associated a priori error estimate. The kernel of the present work is given by Sections 3 and 4, where we develop the residual based a posteriori error analysis. Indeed, in Section 3 we employ a suitable auxiliary problem and apply integration by parts and the local approximation properties of the Cl´ ement interpolant to derive a reliable a posteriori error estimator. In other words, the method that we use to prove reliability combines a technique utilized in mixed finite element schemes with the usual procedure applied to primal finite element methods. It is important to remark that just one of these approaches by itself would not be enough in this case. In addition, up to our knowledge, this combined analysis seems to be applied here for the first time. Next, in Section 4 we make use of inverse inequalities and the localization technique based on triangle-bubble and edge-bubble functions to show that the estimator is efficient. We remark that, because of the new Galerkin least-squares terms employed, most of the residual terms defining the error indicator are new, and hence our proof of efficiency needs to previously establish more general versions of some technical lemmas concerning inverse estimates and piecewise polynomials. Finally, several numerical results confirming reliability, efficiency, and robustness of the estimator with respect to the Poisson ratio, are provided in Section 5. In addition, the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution is also illustrated here.
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27 Lee mas

BOUNDARY CONTROLLABILITY OF A LINEAR SEMI-DISCRETE 1-D WAVE EQUATION DERIVED FROM A MIXED FINITE ELEMENT METHOD

BOUNDARY CONTROLLABILITY OF A LINEAR SEMI-DISCRETE 1-D WAVE EQUATION DERIVED FROM A MIXED FINITE ELEMENT METHOD

Let us briefly explain why the method that we introduce here leads to a uniformly controllable semi-discrete system. As we have mentioned before the main problem from the controllability point of view was pointed out in [8] and it is due to the fact that the velocity of propagation of some high frequency numerical waves may converge to zero when the discretization step, h tends to zero. As a consequence, in order to obtain a uniform controllability result we should consider a controllability time T that tends to infinity as h tends to zero. In our discrete model this phenomenon does not appear. In fact, we show that the speed of propagation of the high frequency oscillations is larger than the one corresponding to the continuous solutions. For the controllability property, this does not present any problem.
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46 Lee mas

Finite element approximation of the vibration problem
			for a Timoshenko curved rod

Finite element approximation of the vibration problem for a Timoshenko curved rod

Remark 2.2. The vibration problem above can be formally obtained from the three- dimensional linear elasticity equations as follows: According to the Timoshenko hypotheses, the admissible displacements at each point ηn + ζb ∈ S (see Fig. 2.1) are of the form u + θ × (ηn + ζb), with u, θ, n and b being functions of the arc-length coordinate s. Test and trial displacements of this form are taken in the variational formulation of the linear elasticity equations for the vibration problem of the three-dimensional rod. By integrating over the cross-sections and multiplying the shear terms by correcting factors k 1 and k 2 , one arrives at problem (2.2).
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14 Lee mas

TítuloStabilized dual mixed method for the problem of linear elasticity with mixed boundary conditions

TítuloStabilized dual mixed method for the problem of linear elasticity with mixed boundary conditions

Mixed finite element methods are typically used in linear elasticity to avoid the effects of locking. They also allow to approximate directly unknowns of physical interest, such as the stresses. We consider here the mixed method of Hellinger and Reissner, that provides simultaneous approxima- tions of the displacement u and the stress tensor σ. The symmetry of the stress tensor prevents the extension of the standard dual-mixed formulation of the Poisson equation to this case. In general, the symmetry of σ is imposed weakly, through the introduction of the rotation as an additional unknown, and stable mixed finite elements for the linear elasticity problem involve many degrees of freedom (see, for instance, [2]).
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6 Lee mas

Greif_Sc...pdf

Greif_Sc...pdf

We introduce a new preconditioning technique for iteratively solving linear systems arising from finite element discretization of the mixed formulation of the time-harmonic Maxwell equations. The precondi- tioners are motivated by spectral equivalence properties of the discrete operators, but are augmentation free and Schur complement free. We provide a complete spectral analysis, and show that the eigenvalues of the preconditioned saddle point matrix are strongly clustered. The analytical observations are accompanied by numerical results that demonstrate the scalability of the proposed approach. Copyright q 2007 John Wiley & Sons, Ltd.
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17 Lee mas

An ADER finite volume method for an atherosclerosis model

An ADER finite volume method for an atherosclerosis model

•We have obtained a numerical solution of a 1D nonlinear reaction-diffusion type model, representing the initial stages of atherosclerosis, which is a variant of the one prop[r]

71 Lee mas

Development of a System for the Computation of Electromagnetic Wave Scattering from Non-Penetrable Objects by Solving the Electric Field Integral Equation

Development of a System for the Computation of Electromagnetic Wave Scattering from Non-Penetrable Objects by Solving the Electric Field Integral Equation

 All the program processing of 2.2 was built on C++, reading the input from gmsh and providing the data output. The compiler used is the GNU GCC compiler for 32 bits windows. The Armadillo library [7] was used for the solution of the system (i.e. solving eq. (12)). This library works on BLAS and LAPACK (libraries made to work with linear algebra), so these libraries had to be also installed. The solver used for the system is an iterative one, i.e. Armadillo does not invert the impedance matrix and then multiply it by the excitation vector, Armadillo guesses a solution for the current vector and performs a multiplication by the impedance matrix; then it calculates the residual trying to make it tend to zero as fast as possible with an algorithm that may be a gradient one. This leads to much faster solution times. A compiled object-oriented language was chosen because of its scalable possibilities against other non-object-oriented, and the fact that it is compiled makes it more efficient, since large and complex problems were intended to be solved.
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56 Lee mas

Implementation of user element subroutines for frequency domain analysis of wave scattering problems with commercial finite element codes

Implementation of user element subroutines for frequency domain analysis of wave scattering problems with commercial finite element codes

The same set of results obtained in the first model is also presented. Figure 3.20 depicts the spatially distributed transfer functions, Figure 3.21 the Fourier spectral amplitude and Figure 3.22 the synthetic seismograms from the three considered methods. The agreement between the results from the full model and the reduced domain method is almost exact, while the results from the classical model exhibit significant differences. The classical model captures only the natural frequency of the microzone, which is in part dominated by the mechanical effect. The classical model however underpredicts the amplitude associated to the natural mode. From the analysis conducted in section 4.1 it is known that near the edges of the canyon large amplifications due to the diffraction effect should be generated. This diffraction effect, which is effectively being captured by the regional model, and subsequently applied to the reduced model seems to be the most relevant aspect of the excitation and the mechanical effect appears to play a minor role in the local response. In a more detailed study a separation of the geometric effects contained in the regional model and the mechanical effects, contained in the micro-zone, should be conducted. That study should provide correction factors to be applied to the standard one-dimensional models to account for the geometric effect.
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89 Lee mas

TítuloA generalized statement for advective diffusive phenomena  Finite
element model and applications

TítuloA generalized statement for advective diffusive phenomena Finite element model and applications

We will derive this formulation by substituting the equation (1.2) (known as Fick’s equation) by a Cattaneo-type law. Cattaneo’s equation involves a tensorial function τ . This mapping transforms each point (x, t) of the domain into that point relaxation tensor . The coordinates of the relaxation tensor are specific diffusion process times. Up to the authors’ knowledge, Cattaneo’s equation has been only used in non-advective thermal problems, see for instance [34]. Thus, we had to find Cattaneo’s equation with convective term [31]. This equation has been derived by using a Lagrangian description but it will be written in this paper in Eulerian coordinates. Hence, basic equations for the transport problem described by using Cattaneo’s law are
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33 Lee mas

Advanced Computational Methods for Dam Protections Against Overtopping

Advanced Computational Methods for Dam Protections Against Overtopping

The most important design aspect of this type of struc- ture is the determination of the leakage flow through the joints. In fact, if this is not correctly defined, the drainage layer placed below the blocks could eventual- ly get saturated and an undesired uplift pressure could compromise the structural stability of the protection. For the above mentioned reasons, CIMNE UPM and PREHORQUISA have been developing a tool for the de- sign of WSBs protection system that will be presented in the next sections. The consultation of the paper “A new evolution on the wedge-shaped block for overtop- ping protection of embankment dams: the ACUÑA block” by F.J. Caballero et al., from this same collection, is recommended for a comprehensive overview of the work performed.
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9 Lee mas

TítuloA validation of the boundary element method for grounding grid design and
computation

TítuloA validation of the boundary element method for grounding grid design and computation

On the other hand, to obtain highly accurate solutions by means of standard techniques (FEM, Finite Dierences) should imply unapproachable computing requirements; analytical solutions ar[r]

11 Lee mas

TítuloOn the resolution of the viscous incompressible flow for various SUPG finite
element formulations

TítuloOn the resolution of the viscous incompressible flow for various SUPG finite element formulations

The ‘cheapest’ storing mechanism is to keep exclusively those elements different from zero. This is a much better procedure that avoids wasting memory resources in storing mid- height zeros, which can be more numerous than the number of non-zero elements even when the mesh is re-numbered so as to reduce the band width to a minimum and specially when a mixed method is used. Provided that the sparse storage cannot be used in combination with a direct solver because some elements could be ‘thrown out’ of the sparse stencil, when this type of storage is used, some other algorithm should be implemented to solve the system of equations. For the present calculations a Preconditioned Biconjugate Gradient Method (PBCG) type of solver will be implemented in order to solve the resulting system.
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20 Lee mas

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