We present and analyse a new mixedfiniteelementmethod 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 finiteelement 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 finiteelement solutions. Several numerical results illustrate the performance of the method and its capability to localize boundary layers, inner layers, and singularities.
The most popular approach in applications is based on the mixed formulation, with pressure and velocity as unknowns. It is well-known that the Galerkin scheme associated to this formulation is not always well-posed and stability is ensured only for certain combinations of finiteelement subspaces. In this framework, several stabilization methods have been proposed in the literature. We consider a stabilized mixedfiniteelementmethod introduced by Masud and Hughes in  for isotropic porous media. This method is based on the addition of suitable residual type terms to the standard dual-mixed approach. The resulting scheme is stable for any combination of continuous velocity and pressure interpolations, and has the singularity that the stabilization parameters can be chosen independently of the mesh size. This property was already present in the modified mixed formulation introduced in  for second order elliptic problems. The stabilization introduced in  was also applied in  to analyze a mixed discontinuous Galerkin method for Darcy flow. A similar idea is used in  to derive several unconditionally stable mixedfiniteelement methods for Darcy flow. Finally, concerning the a posteriori error analysis of the method proposed in , a residual based a posteriori error estimate of the velocity in L 2 -norm was derived in .
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  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.
In general, any semi-discrete dynamics generates spurious high-frequency oscillations that do not exist at the continuous level. Moreover, a dispersion phenomenon appears and the velocity of propagation of these high frequency numerical waves may converge to zero when the mesh size tends to zero. Note that these spurious oscillations correspond to the high frequencies of the discrete model and therefore, they weakly converge to zero when the discretization parameter h does. Consequently, their existence is compatible with the convergence of the numerical scheme. However, when we are dealing with the exact controllability problem, an uniform time for the control of all numerical waves is needed. Since the velocity of propagation of some high frequency numerical waves may tend to zero with the mesh size, the uniform controllability properties of the semi-discrete model may eventually disappear for a fixed time T > 0. This is the case when the semi-discrete model is obtained by finite differences or the classical finiteelementmethod (see , for a detailed analysis of the 1-D case and  for the 2-D case, in the context of the dual observability problem).
Now, before presenting the examples, we would like to remark in advance that, as compared with more traditional mixed methods, and besides the fact, already emphasized, of being able to choose any ﬁnite element subspace, our augmented approach presents other important advantages, as well. Indeed, let us ﬁrst observe that in the case of uniform reﬁnements each interior edge (resp. interior node) belongs to 2 (resp. 6) triangles, which yields corresponding correction factors of 1 2 and 1 6 when counting the global number of degrees of freedom, say N , in terms of the number of triangles, say M . Then, it is not diﬃcult to see that the number of unknowns N of (10) behaves asymptotically as 5 M , whereas this behaviour is given by 7 . 5 M when the well-known PEERS from  is used in the Galerkin scheme of the non-augmented formulation. In other words, the discrete system using PEERS introduces about 50% more degrees of freedom than our approach at each mesh, and therefore the augmented method becomes a much cheaper alternative. Furthermore, it is important to note that the polynomial degrees involved in the deﬁnition of H h σ × H u
This is the second part of a work dealing with a low-order mixed ﬁnite elementmethod for a class of nonlinear Stokes models arising in quasi-Newtonian ﬂuids. In the ﬁrst part we showed that the resulting variational formulation is given by a twofold saddle point operator equation, and that the corresponding Galerkin scheme becomes well posed with piecewise constant functions and Raviart–Thomas spaces of lowest order as the associated ﬁnite element sub- spaces. In this paper we develop a Bank–Weiser type a posteriori error analysis yielding a reliable estimate and propose the corresponding adaptive algorithm to compute the mixed ﬁnite element solutions. Several numerical results illus- trating the eﬃciency of the method are also provided.
Abstract. The aim of this paper is to analyze a mixedfiniteelementmethod for computing the vibration modes of a Timoshenko curved rod with arbitrary geometry. Optimal order error estimates are proved for displacements and rotations of the vibration modes, as well as a double order of convergence for the vibration frequencies. These estimates are essentially independent of the thickness of the rod, which leads to the conclusion that the method is locking free. A numerical test is reported in order to assess the performance of the method.
The application of stabilization techniques allows to use simpler finiteelement subspaces, includ- ing convenient equal-order interpolations that are generally unstable within the mixed approach. Recently, a new stabilized mixedfiniteelementmethod was presented in  for the problem of linear elasticity in the plane. This method leads to a well-posed, locking-free Galerkin scheme for any choice of finiteelement subspaces when homogeneous Dirichlet boundary conditions are prescribed. Moreover, in the simplest case it requires less degrees of freedom than the classical PEERS or BDM. The method was successfully extended in  to the case of non-homogeneous Dirichlet boundary conditions; the three-dimensional version can be found in .
An Uzawa-type algorithm for the saddle point system, coupled with a domain decomposition approach, has been proposed in [ 12 ] . The original system is transformed into a new system by augmentation with the scalar Laplacian as a weight matrix, and it is shown that the condition number of the resulting preconditioned system grows logarithmically with respect to the ratio between the subdomain diameter and the mesh size. The method incorporates augmentation and is parameter dependent. Its convergence properties rely on extreme eigenvalues of the augmented Schur complement, which may be difficult to evaluate.
It is recognised that low frequency sound transmission into and between dwellings is an increasing contribution to nuisance. Sources of low frequency noise include hi-fi and home movie systems of high power and enhanced bass response, domestic appliances and road traffic. If heavyweight walls and floors and lightweight cavity partitions fail insulation standards, they tend to do so at low frequencies. Current standards [1,2] deal only with the frequency range 100-3150 Hz and generally there is poor reproducibility between laboratories below this frequency range and a method of measurement of insulation below 100 Hz has yet to be agreed . A fundamental question remains on how laboratory measurements can be related to field performance of the party wall, floor or partition when installed. Diffuse or neutral field conditions in test chambers do not correspond to the highly modal characteristics at low frequencies and for small room volumes associated with dwellings .
This system of equations are solved by use of an incremental formulation. Here ˆ u and ˆ p are the vectors of nodal point incremental displacements and pressures, respectively. t+∆t R is the vector of externally applied nodal point loads corresponding to time t + ∆t; and F = t FU, t FP T is the vector of nodal point forces corresponding to the internal element stresses at time t. Due to this, the subtraction at the right hand side is known as the out-of-balance force vector and may reach an small value at the end of each load increment. Generally, this value is established as a percentage of the first out-of-balance vector magnitude at a load increment.
First of all, I want to thank my thesis director Ana Laverón Simavilla for the op- portunity of doing my Ph.D in this subject and for all the advise and support I got from her, not only for this thesis development but also for my professional skills evolution. In the same line, I want to thank my thesis director Victoria Lapuerta González for all the effort invested in remembering what those panels where about, that I know was hard sometimes but nice some others. Everything started for me with some optional lessons at the university about numerical aerodynamics that Ana and Vicky used to teach and that I loved to teach, although the work on this panel method subject started a lot of time before me.
T h e use of several k i n d s of elements within the family c a n relieve s o m e c o m p u t a t i o n a l effort. L o w degree elements can be used near t h e b o u n d a r y , n o r m a l l y in large n u m b e r s in o r d e r t o m o d e l the geometry, a n d o n l y a few high degree elements a r e usually required in the central a r e a of t h e plate. O b v i o u s l y t h e hierarchic family s h o u l d include t r a n s i t i o n a l elements. T h e s e can be o b t a i n e d either directly or f r o m a n o r m a l element by r e d u c t i o n of t h e o r d e r a l o n g s o m e sides of t h e element.
The finiteelementmethod (FEM) has developed rapidly as a topic ofresearch in electrical engineering. It is a numerical method that can be used to calculate the electromagnetic fields, electromagnetic forces, energy storage in electromagnetic fields, power dissipation, eddy current, thermal conduction etc. Calculation ofthe transmission line impedance using the FEM requires that an appropriate boundary be defined first. Ali bodies (including the air and the ground) included in the boundary are modeled as areas. Each area has its own geo metric shape and physical properties, such as conductivity, permeability, nonlinearity ofmagnetization etc. Ali areas are subdivided into first order triangular finite elements. The boundary conditions must then be given to allow Maxwell's equations to be solved correctly according to the nature ofthe problem.
In this project, the Method of Moments has been thoroughly reviewed, starting from the mathematical base, then applying the method to an electrostatic problem (the capacitor, where two of its characteristics were studied: capacity and charge distribution), which was programmed using MATLAB. Up to this point, the potential of a program with these characteristics can be noticed. It is a program that can provide accurate results and plenty information about the behavior of the object once it has been solved. Finally, the MoM has been programmed for the solution of electromagnetic wave scattering in non-penetrable PEC objects of arbitrary shape, allowing the determination of the radar cross section of those objects.
As a first approximation for 2D case, Eq. (17) can be solved as an infinite waveguide where the refraction index is supposed constant along the propagation axis ( z -direction). Here, the electric field can be computed by FEM (See more details in [32-34]). To calculate the field by FEM, the fiber cross-section is discretized into small elements and Eq. (17) is computed for each element (See Fig. 1(b) and (c)). After calculating the electric field E for each node, and for the M -modes supported by a SMMF, M 2 a NA 2 / 2 with a
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.
continuity requierements, including the fir9t deri vative or even the curvature. The most important aspect of the FEM both for this type of problem snd for other problems, involves the selection of shape or interpolation functions, since the comp£ sing and solving of the system of equations (11) is standard and can be found in any general matrix programmes of structures, for example: SAP, STRUDL, ANSYS, NASTRAN, etc. Figure B shows the possibili- ty of C 2 element which might be used for this type of problem of smooth and continous surface repre- sentation. If only the continuity of the slope is required, the selection of triangular and compat_i ble elements is very wide, and these elements co- rrespond to all the compatible elements of bending of plates (refer to the specialised literature C A S E 6 7 8
element was composed of 20 contact points. The elements of the Komazume and Doza parts were made smaller than the other parts. The number of circumferential divisions was 60, and the upper two small areas were selected as fixed areas, as shown in the figure. The analysis software is Marc K7.3.