By using the finite element $p$-Version in convection-diffusion problems, we can attain to a stabilized and accurate results. Furthermore, the fundamental of the finite element $p$-Version is augmentation degrees of freedom. Based on the fact that the finite element and the meshless methods have similar concept, it is obvious that many ideas in the finite element can be easily used in the meshl...

We propose an extension of the discontinuous Galerkin local orthogonal decomposition multiscale method, presented in [14], to convection-diffusion problems with rough, heterogeneous, and highly varying coefficients. The properties of the multiscale method and the discontinuous Galerkin method allows us to better cope with multiscale features as well as interior/boundary layers in the solution. ...

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Numerical solution of convection-dominated problems requires the use of layer-adapted anisotropic meshes. Since a priori construction of such meshes is difficult for complex problems, it is proposed to generate them in an adaptive way by moving the node positions in the mesh such that an a posteriori error estimator of the overall error of the approximate solution is reduced. This approach is f...

The cell-vertex formulation of the finite volume method has been developed and widely used to model inviscid flows in aerodynamics: more recently, one of us has proposed an extension for viscous flows. The purpose of the present paper is two-fold: first we have applied this scheme to a well-known convection-diftusion model problem, involving flow round a 180° bend, which highlights some of the ...

In this paper we derive an a posteriori error bound for the Lagrange–Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Based on th...

We survey recent developments and give some new results concerning uniqueness of weak and renormalized solutions for degenerate parabolic problems of the form ut − div (a0(∇w) + F (w)) = f , u ∈ β(w) for a maximal monotone graph β, a Leray-Lions type nonlinearity a0, a continuous convection flux F , and an initial condition u|t=0 = u0. The main difficulty lies in taking boundary conditions into...