There are many methods and approaches to solving convection--diffusion problems. For those who want to solve such problems the situation is very confusing and it is very difficult to choose the right method. The aim of this short overview is to provide basic guidelines and to mention the common features of different methods. We place particular emphasis on the concept of linear and non-linear stabilization and its implementation within different approaches.

We study in this paper some systems, using standard tools devoted to the analysis of semilinear elliptic problems on R3. These systems do not admit any non trivial radial solutions in the E1 E2 = + 1 cases. A first type of solution consists in a ground state of R (-1,-1), exhibited by variational arguments, whose structure is a finite energy perturbation of a non trivial constant solution of R (-1,-1). A second type consists in a radial, oscillating, asymptotically null at infinity solution in the...

In this paper, some superconvergence results of high-degree finite element method are obtained for solving a second order elliptic equation with variable coefficients on the inner locally symmetric mesh with respect to a point x 0 for triangular meshes. By using of the weak estimates and local symmetric technique, we obtain improved discretization errors of O(h p+1 |ln h|2) and O(h p+2 |ln h|2) when p (≥ 3) is odd and p (≥ 4) is even, respectively. Meanwhile, the results show that the combination...

We develop the analysis of stabilized sparse tensor-product finite element methods for high-dimensional, non-self-adjoint and possibly degenerate second-order partial differential equations of the form $-a:\nabla \nabla u+b·\nabla u+cu=f\left(x\right)$, $x\in \Omega ={(0,1)}^d\subset {ℝ}^d$, where $a\in {ℝ}^{d\times d}$ is a symmetric positive semidefinite matrix, using piecewise polynomials of degree p ≥ 1. Our convergence analysis is based on new high-dimensional approximation results in sparse tensor-product spaces. We show that the error between the analytical solution u and its stabilized sparse...

Let $AV\to V^{\text{'}}$ be a strongly elliptic operator on a $d$-dimensional manifold $D$ (polyhedra or boundaries of polyhedra are also allowed). An operator equation $Au=f$ with stochastic data $f$ is considered. The goal of the computation is the mean field and higher moments ${ℳ}^{1}u\in V$, ${ℳ}^{2}u\in V\otimes V$, $...$, ${ℳ}^{k}u\in V\otimes \cdots \otimes V$ of the solution. We discretize the mean field problem using a FEM with hierarchical basis and $N$ degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment ${ℳ}^{k}u$ for $k\ge 1$. The key tool...

We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore...

Special exact curved finite elements useful for solving contact problems of the second order in domains boundaries of which consist of a finite number of circular ares and a finite number of line segments are introduced and the interpolation estimates are proved.

We consider a model coupling the Darcy equations in a porous medium with the Navier-Stokes equations in the cracks, for which the coupling is provided by the pressure's continuity on the interface. We discretize the coupled problem by the spectral element method combined with a nonoverlapping domain decomposition method. We prove the existence of solution for the discrete problem and establish an error estimation. We conclude with some numerical tests confirming the results of our analysis.

By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec’s edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proof is new, instead, and shows how families of spurious-free elements can...

By using an inductive procedure we prove that the Galerkin finite element approximations of electromagnetic eigenproblems modelling cavity resonators by elements of any fixed order of either Nedelec's edge element family on tetrahedral meshes are convergent and free of spurious solutions. This result is not new but is proved under weaker hypotheses, which are fulfilled in most of engineering applications. The method of the proof is new, instead, and shows how families of spurious-free elements...

We consider some abstract nonlinear equations in a separable Hilbert space $H$ and some class of approximate equations on closed linear subspaces of $H$. The main result concerns stability with respect to the approximation of the space $H$. We prove that, generically, the set of all solutions of the exact equation is the limit in the sense of the Hausdorff metric over $H$ of the sets of approximate solutions, over some filterbase on the family of all closed linear subspaces of $H$. The abstract results are...

We consider numerical approximations of stationary incompressible Navier-Stokes flows in 3D exterior domains, with nonzero velocity at infinity. It is shown that a P1-P1 stabilized finite element method proposed by C. Rebollo: A term by term stabilization algorithm for finite element solution of incompressible flow problems, Numer. Math. 79 (1998), 283–319, is stable when applied to a Navier-Stokes flow in a truncated exterior domain with a pointwise boundary condition on the artificial boundary....