First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems

We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data f in L2 space. The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov - Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014), pp. 537-552]. This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods - numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform

Stabilization arising from PGEM : a review and further developments

The aim of this paper is twofold. First, we review the recent Petrov-Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP's in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising in a natural way after performing static condensation. The resulting stabilized method is shown to lead to optimal convergences, and afterward, it is numerically validated

On the natural stabilization of convection dominated problems using high order Bubnov–Galerkin finite elements

In the case of dominating convection, standard Bubnov–Galerkin finite elements are known to deliver oscillating discrete solutions for the convection–diffusion equation. This paper demonstrates that increasing the polynomial degree (p-extension) limits these artificial numerical oscillations. This is contrary to a widespread notion that an increase of the polynomial degree destabilizes the discrete solution. This treatise also provides explicit expressions as to which polynomial degree is sufficiently high to obtain stable solutions for a given Péclet number at the nodes of a mesh

Stabilized finite element methods for time dependent convection-diffusion equations

Thesis (Doctoral)--Izmir Institute of Technology, Mathematics, Izmir, 2012Includes bibliographical references (leaves: 92-96)Text in English; Abstract: Turkish and Englishx, 96 leavesIn this thesis, enriched finite element methods are presented for both steady and unsteady convection diffusion equations. For the unsteady case, we follow the method of lines approach that consists of first discretizing in space and then use some time integrator to solve the resulting system of ordinary differential equation. Discretization in time is performed by the generalized Euler finite difference scheme, while for the space discretization the streamline upwind Petrov-Galerkin (SUPG), the Residual free bubble (RFB), the more recent multiscale (MS) and specific combination of RFB with MS (MIX) methods are considered. To apply the RFB and the MS methods, the steady local problem, which is as complicated as the original steady equation, should be solved in each element. That requirement makes these methods quite expensive especially for two dimensional problems. In order to overcome that drawback the pseudo approximation techniques, which employ only a few nodes in each element, are used. Next, for the unsteady problem a proper adaptation recipe, including these approximations combined with the generalized Euler time discretization, is described. For piecewise linear finite element discretization on triangular grid, the SUPG method is used. Then we derive an efficient stability parameter by examining the relation of the RFB and the SUPG methods. Stability and convergence analysis of the SUPG method applied to the unsteady problem is obtained by extending the Burman’s analysis techniques for the pure convection problem. We also suggest a novel operator splitting strategy for the transport equations with nonlinear reaction term. As a result two subproblems are obtained. One of which we may apply using the SUPG stabilization while the other equation can be solved analytically. Lastly, numerical experiments are presented to illustrate the good performance of the method

Pseudo residual-gree bubble functions for the stabilization of convection-diffusion-reaction prolems

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2012Includes bibliographical references (leaves: 55-57)Text in English; Abstract: Turkish and Englishx, 57 leavesConvection - diffusion - reaction problems may contain thin regions in which the solution varies abruptly. The plain Galerkin method may not work for such problems on reasonable discretizations, producing non-physical oscillations. The Residual - Free Bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless in special limit cases. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. In my thesis we will examine a stabilization technique, based on the RFB method and particularly designed to treat the most interesting case of small diffusion in one and two space dimensions for both steady and unsteady convection - diffusion - reaction problems. We replace the RFB functions by their cheap, but efficient approximations which retain the same qualitative behavior. We compare the method with other stabilized methods

Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

It is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the SUPG-type for the convection-diffusion-reaction problems. In particular, the residual-free bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless the configuration is simple. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. Here we propose a stabilization technique based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to L 1 norm. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. This fact is also confirmed by numerical experiments. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one in Brezzi et al. (Math. Models Methods Appl. Sci. 13:445-461, 2003). © 2011 Springer-Verlag

A priori and a posteriori analysis of non-conforming finite elements with face penalty for advection-diffusion equations

We analyse a non-conforming finite-element method to approximate advection-diffusion-reaction equations. The method is stabilized by penalizing the jumps of the solution and those of its advective derivative across mesh interfaces. The a priori error analysis leads to (quasi-)optimal estimates in the mesh size (sub-optimal by order ½ in the L2-norm and optimal in the broken graph norm for quasi-uniform meshes) keeping the Péclet number fixed. Then, we investigate a residual a posteriori error estimator for the method. The estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. Finally, to illustrate the theory we present numerical results including adaptively generated meshe