We discuss the finite element approximation of eigenvalue problems associated with compact operators. While main emphasis is on symmetric problems, some comments are present for non-self-adjoint operators as well. The topics covered include standard Galerkin approximations, non-conforming and in mixed form. Some applications theory presented and, particular, Maxwell problem discussed detail. final part tries to introduce reader fascinating setting differential forms homological techniques...

The purpose of this paper is to address some difficulties which arise in computing the eigenvalues Maxwell's system by a finite element method. Depending on method used, spectrum may be polluted spurious modes are difficult pick out among approximations physically correct eigenvalues. Here we propose criterion establish whether or not scheme well suited approximate eigensolutions and, positive case, estimate rate convergence eigensolutions. This involves properties space and suitable Fortin...

We consider the approximation properties of quadrilateral finite element spaces vector fields defined by Piola transform, extending results previously obtained for scalar approximation. The are constructed starting with a given dimensional space on square reference element, which is then transformed to each convex via transform associated bilinear isomorphism onto element. For affine isomorphisms, necessary and sufficient condition order r + 1 in L2 that component functions contain all...

We consider the approximation properties of finite element spaces on quadrilateral meshes. The are constructed starting with a given dimensional space functions square reference element, which is then transformed to each convex via bilinear isomorphism onto element. It known that for affine isomorphisms, necessary and sufficient condition order $r+1$ in $L^p$ $r$ $W^1_p$ contain all polynomial total degree at most $r$. In case it same estimates hold if function contains separate show, by...

In continuum mechanics problems, we have to work in most cases with symmetric tensors,symmetry expressing the conservation of angular momentum. Discretization symmetrictensors is however difficult and a classical solution employ some form reduced symmetry.We present two ways introducing elements symmetry.The first one based on Stokes two-dimensional case allows torecover practically all interesting market. This is(definitely) not true three dimensions. On other hand second approach (based...

In the approximation of linear elliptic operators in mixed form, it is well known that so-called inf-sup and ellipticity kernel properties are sufficient (and, a sense to be made precise, necessary) order have good optimal error bounds. One might think, spirit Mercier-Osborn-Rappaz-Raviart consideration behavior commonly used elements (like Raviart–Thomas or Brezzi–Douglas–Marini elements), these conditions also ensure convergence for eigenvalues. this paper we show not case. particular...

We introduce a new formulation for the finite element immersed boundary method which makes use of distributed Lagrange multiplier. prove that full discretization our model, based on semi-implicit time advancing scheme, is unconditionally stable with respect to step size.

In this work, we introduce a novel algorithm for the Biot problem based on hybrid high-order discretization of mechanics and symmetric weighted interior penalty flow. The method has several assets, including, in particular, support general polyhedral meshes arbitrary space approximation order. Our analysis delivers stability error estimates that hold also when specific storage coefficient vanishes, shows constants have only mild dependence heterogeneity permeability coefficient. Numerical...

The tetrahedral finite element approximation of the Stokes problem is analyzed by means polynomials piecewise degree k+1 for velocity and continuous k pressure. A stability result given every $k\ge 1$.

The immersed boundary method is both a mathematical formulation and numerical method. In its continuous version it fully nonlinearly coupled for the study of fluid structure interactions. Many methods have been introduced to reduce difficulties related nonlinear coupling between evolution. However instabilities arise when explicit or semi-implicit are considered. this work we present stability analysis based on energy estimates variational A two-dimensional incompressible in form simple...

We prove the stability for approximation of stationary Stokes equations by means piecewise continuous velocities degree k+1 and pressures k k≥1. The necessary sufficient condition required on triangulation is that it contains at least three triangles. theorem compared with previous results.

The Immersed Boundary Method (IBM) has been designed by Peskin for the modeling and numerical approximation of fluid-structure interaction problems, where flexible structures are immersed in a fluid. In this approach, Navier–Stokes equations considered everywhere presence structure is taken into account means source term which depends on unknown position structure. These coupled with condition that moves at same velocity underlying Recently, finite element version IBM developed, offers...