It was shown two decades ago that the Pk-Pk−1 mixed element on triangular grids, approximating the velocity by the continuous Pk piecewise polynomials and the pressure by the discontinuous Pk−1 piecewise polynomials, is stable for all k ≥ 4, provided the grids are free of a nearly-singular vertex. The problem with the method in 3D was posted then and remains open. The problem is solved partiall...

Over the last few years, several mixed discontinuous Galerkin finite element methods (DGFEM) have been proposed for the discretization of incompressible fluid flow problems. We mention here only the piecewise solenoidal discontinuous Galerkin methods introduced in [5,25], the local discontinuous Galerkin methods of [12,11], and the interior penalty methods studied in [24,33,18]. Some of the mai...

Amongst the more exciting phenomena in the field of nonlinear partial differential equations is the Lavrentiev phenomenon which occurs in the calculus of variations. We prove that a conforming finite element method fails if and only if the Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods have to be designed for the detection of the Lavrentiev phenomenon in the ...

where gh is an approximation of g and V 0 h ⊂ H 1 0 (Ω) is a finite element subspace. In practice, gh is considered to be a nodal interpolation of g in Vh|∂Ω, where Vh ⊂ H(Ω) is a finite element subspace. However it is shown in [4] that the error estimates for (3) are better when gh is chosen to be the L2-projection of g onto Vh|∂Ω. In particular when gh is chosen to be a nodal interpolation, t...

The present paper proposes and analyzes an interior penalty technique using C0-finite elements to solve the Maxwell equations in domains with heterogeneous properties. The convergence analysis for the boundary value problem and the eigenvalue problem is done assuming only minimal regularity in Lipschitz domains. The method is shown to converge for any polynomial degrees and to be spectrally cor...

The paper presents an algorithm of adaptation by successive mesh regeneration and its application to the potential nonlinear flow problems. The Newton-Raphson method is applied to the solution of nonlinear problem. The Kutta-Zhukovsky condition is implemented by the penalty function method. The adaptation algorithm is based on the modification of mesh size function depending on the error indica...

In this talk we will discuss several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty (WOPSIP) method. The WOPSIP method belongs to the family of discontinuous finite element methods. Theoretical results on the condition number estimate of the preconditioned system will be presented along with numerical results. THE DISCRETE PROBLEM Let Ω be a bo...

The paper demonstrates that enhanced stability properties of some finite element methods on barycenter refined meshes enables efficient numerical treatment of problems involving incompressible or nearly incompressible media. One example is the linear elasticity problem in a pure displacement formulation, where a lower order finite element method is studied which is optimal order accurate and ro...