1,721,058 research outputs found
Superconvergence of the MINI mixed finite element discretization of the Stokes problem: An experimental study in 3D
No full textStokes flows are a type of fluid flow where convective forces are small in comparison with viscous forces, and momentum transport is entirely due to viscous diffusion. Besides being routinely used as benchmark test cases in numerical fluid dynamics, Stokes flows are relevant in several applications in science and engineering including porous media flow, biological flows, microfluidics, microrobotics, and hydrodynamic lubrication. The present study concerns the discretization of the equations of motion of Stokes flows in three dimensions utilizing the MINI mixed finite element, focusing on the superconvergence of the method which was investigated with numerical experiments using five purpose-made benchmark test cases with analytical solution. Despite the fact that the MINI element is only linearly convergent according to standard mixed finite element theory, a recent theoretical development proves that, for structured meshes in two dimensions, the pressure superconverges with order O(h3/2), as well as the linear part of the computed velocity with respect to the piecewise-linear nodal interpolation of the exact velocity. The numerical experiments documented herein suggest a more general validity of the superconvergence in pressure, possibly to unstructured tetrahedral meshes and even up to quadratic convergence which was observed with one test problem, thereby indicating that there is scope to further extend the available theoretical results on convergence
A reduced order model for the finite element approximation of eigenvalue problems
No full textIn this paper we consider a reduced order method for the approximation of the eigensolutions of the Laplace problem with Dirichlet boundary condition. We use a time continuation technique that consists in the introduction of a fictitious time parameter. We use a POD approach and we present some theoretical results showing how to choose the optimal dimension of the POD basis. The results of our computations confirm the optimal behavior of our approximate solution. We compute the first eigenvalue and discuss how to approximate the next eigenmodes
An alternative approach to the analysis and the approximation of the Navier-Stokes equations
No full textA suitable restatement of the Navier-Stokes equations provides a new tool for their numerical approximation. Spectral methods are used in the experiments to show the effectiveness of the technique. The main achievement is the total elimination of spurious modes
Stability and Geometric Conservation Laws for ALE formulations
No full textThe aim of this paper is to investigate a model ALE scheme, with respect to various possible choices of time discretizations. For each time scheme, we investigate the relationships between stability and the so-called Geometric Conservation Laws (CGL). We shall see that GCL condition proves neither necessary nor sufficient for stability. In doing so, we review some known theoretical results and we prove some new stability results for space-time ALE discretizations. Some new error estimates are also presented. Several numerical experiments confirm the theory
Iterative ilu preconditioners for linear systems and eigenproblems
No full textIterative ILU factorizations are constructed, analyzed and applied as preconditioners to solve both linear systems and eigenproblems. The computational kernels of these novel Iterative ILU factorizations are sparse matrix-matrix multiplications, which are easy and efficient to implement on both serial and parallel computer architectures and can take full advantage of existing matrix-matrix multiplication codes. We also introduce level-based and threshold-based algorithms in order to enhance the accuracy of the proposed Iterative ILU factorizations. The results of several numerical experiments illustrate the efficiency of the proposed preconditioners to solve both linear systems and eigenvalue problems
Analysis of finite element approximation of evolution problems in mixed form
No full textThis paper deals with the finite element approximation of evolution problems in mixed form. Following [D. Boffi, F. Brezzi, L. Gastaldi, Math. Comp. 69 (2000), pp.121-140], we handle separately two types of problems. A model for the first case is the heat equation in mixed form, while the time dependent Stokes problem fits within the second one. For either case, we give sufficient conditions for a good approximation in the natural functional spaces. The results are not obvious in the first situation. In this case, the well-known conditions for the well-posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem. This issue is demonstrated with a numerical (counter-) example and justified analytically
On the Spectrum of an Operator Associated with Least-Squares Finite Elements for Linear Elasticity
No full textIn this paper we provide some more details on the numerical analysis and we present some enlightening numerical results related to the spectrum of a finite element least-squares approximation of the linear elasticity formulation introduced recently. We show that, although the formulation is robust in the incompressible limit for the source problem, its spectrum is strongly dependent on the Lamé parameters and on the underlying mesh
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