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This work presents a numerical solution of the two-dimensional diffusion equation in comparison with the analytical solution. The norms L2 and L∞ of the error are evaluated for two variants of the finite element method: the Galerkin Finite Element Method (GFEM) and the Least-Squares Finite Element Method (LSFEM). Two applications are presented and...
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... this case a purely diffusive problem with source term and boundary conditions of first and second kind has been analyzed. Now at the contours x = 0 and y = 0 the fluxes are specified as shown in Figure 6. In this case another difference is that the meshes are of linear and quadratic triangular elements of size h varying from 2, 82 × 10 −1 to 3, 54 × 10 −2 . ...
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Citations
... However most part of the Mechanics of Fluids problems and Heat and Mass Transfer are classified as dominant convective. Several authors presented papers in the last decades demonstrating that the application of the Galerkin Method in the solution of dominant convective problems in transient regime generate numerical oscillations, for example, Camprub et al. (2000), Romão (2004) e Romão et al. (2008), fact that could be avoided only with a large refinement of the mesh what would be computationally and financially expensive for bi and three-dimensional problems. ...
Convection dominated flows result in a hyperbolic system of equations which leads to ill-conditioned matrices and oscillatory approximations when using the classical Galerkin Finite Element Method (GFEM). In this paper, the Least Square Finite Method (LSFEM) is introduced in the study of transient bidimensional convection-diffusion problems. The differentiated equation of second order which describes the convective-diffusive phenomenon is transformed into an equivalent system of partial differentiated equations of first order which is discretized by the formulation of the LSFEM resulting in a defined algebraic, symmetrical and positive system. The performance of the method is verified by the solution of a test- problem.
... In the last decades, heat transfer and fluid mechanics problems have been analyzed in terms of different numerical schemes. Of these, the Finite Element Method retains flexibility and ease in dealing with complex domains, which, to be divided often generates a non-ordered matrix causing a high computational time (see [2]; [4]), and the high computational cost caused by the calculation of integrals that generate the coefficients of the main matrix (see [5]; [6]). Now, the Finite Difference Method replace the partial derivatives that appear in the governing equations by the algebraic difference quotients yielding an ordered matrix which can be resolved by any linear system solution method, using only non-zero coefficients of the matrix, usually highly sparse (see [7]). ...
This paper aims to extend the Taylor series obtaining expressions for the discretization of the derivatives of first and second order accurate O(Δx 6 ) using central differences. The convection-diffusion-reaction (CDR) equation with variable coefficients and Robin boundary coriditions is chosen, aiming several numerical comparisons and analysis of some special points in the computational domain, the derivative of the first order (convective term) is extended through the backward and forward differences both O(Δx 3 ). Two numerical applications with analytical solution using the L ∞ norm validate the computational code and facilitate the analysis of results.
... The GFEM applied for this type of problems generally produces oscillating solutions for high Péclet and Reynolds numbers, but it has a low computational cost when compared with the PGFEM and LSFEM methods. Recently, several authors have presented applications of the finite element method for two and tridimensional problems, among them [7][8][9][10]. ...
This paper presents the numerical solution, by the Galerkin and Least Squares Finite Element Methods, of the three-dimensional Poisson and Helmholtz equations, representing heat diffusion in solids. For the two applications proposed, the analytical solutions found in the literature review were used to compare with the numerical solutions. The analysis of results was made from the L2 norm (average error throughout the domain) and L∞ norm (maximum error in the entire domain). The results of the two applications (Poisson and Helmholtz equations) are presented and discussed for testing of the efficiency of the methods.
