We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to… (More)
Since the early 70's, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and their use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay… (More)
We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element , which is then transformed to a space of functions on each convex quadri-lateral element via a bilinear isomorphism of the square onto… (More)
In the approximation of linear elliptic operators in mixed form, it is well known that the so-called inf-sup and ellipticity in the kernel properties are sufficient (and, in a sense to be made precise, necessary) in order to have good approximation properties and optimal error bounds. One might think, in the spirit of Mercier-Osborn-Rappaz-Raviart and in… (More)
The immersed boundary method is both a mathematical formulation and a numerical method. In its continuous version it is a fully non-linearly coupled formulation for the study of fluid structure interactions. As it is common in these cases, many numerical methods have been introduced to reduce the difficulties related to the non-linear coupling between the… (More)
The immersed boundary (IB) method is both a mathematical formulation and a numerical method for fluid-structure interaction problems, where immersed incompressible visco-elastic bodies or boundaries interact with an incompressible fluid. Previous formulations of the IB method were not able to treat appropriately thick materials modeled by hyper-elastic… (More)
The tetrahedral finite element approximation of the Stokes problem is analyzed by means of polynomials piecewise of degree k + 1 for the velocity and continuous piecewise of degree k for the pressure. A stability result is given for every k ≥ 1.
Recent results prove that Nédélec edge elements do not achieve optimal rate of approximation on general quadrilateral meshes. In particular, lowest order edge elements provide stable but non convergent approximation of Maxwell's eigenvalues. In this paper we analyze a modification of standard edge element that restores the optimality of the convergence.… (More)