Juhani Pitkäranta

FINITE ELEMENT METHODS FOR LINEAR HYPERBOLIC PROBLEMS

Abstract We give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first-order linear hyperbolic problems.

An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation

We prove Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain L2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. Lp estimates for p * 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.

Direct and inverse error estimates for finite elements with mesh refinements

The finite element method is used to solve a second order elliptic boundary value problem on a polygonal domain. Mesh refinements and weighted Besov spaces are used to obtain optimal error estimates and inverse theorems.

Efficient preconditioning for the p -version finite element method in two dimensions

Parallel preconditioners are formulated and analyzed for systems of equations arising from the p-version finite element method applied to second-order self adjoint elliptic boundary value problems in two dimensions. By using new theoretical results for polynomial spaces, it is proved that the condition number of the preconditioned system grows as

Analysis of mixed methods using mesh dependent norms

\log ^2 p

The problem of membrane locking in finite element analysis of cylindrical shells

, where p is the degree of the polynomial space. Numerical results are presented showing that the condition number indeed grows very slowly with p.

Shell deformation states and the finite element method: A benchmark study of cylindrical shells

Abstract : This paper presents a new approach to the analysis of mixed methods for the approximate solution of 4th order elliptic boundary value problems. In this approach one introduces a pair of mesh dependent norms and proves the approximation method is stable with respect to these norms. The error estimates then follow in a direct manner. In a mixed method, one introduces an auxiliary variable, usually representing another physically important quantity, and writes the differential equation as a lower order system. One then considers Ritz-Galerkin approximation schemes based on a variational formulation of this lower order system, thereby obtaining direct approximations to both the original and auxiliary variables. Three particular mixed methods for the approximate solution of the biharmonic problem are examined in detail.

Analysis of some low-order finite element schemes for Mindlin-Reissner and Kirchhoff plates

SummaryWe analyse the problem of membrane locking in (h, p) finite element models of a thin hemicylindrical shell roof loaded by a smoothly varying normal pressure distribution. We show that in the standard finite element method, locking occurs especially at low values ofp and when the finite element grid is not aligned with the axis of the cylinder. A general strategy of avoiding locking by using modified bilinear forms is introduced, and a special implementation of this strategy on aligned rectangular grids is considered.

Boundary subspaces for the finite element method with Lagrange multipliers

Abstract We demonstrate some of the basic mathematical characteristics of shell deformation states and shell models from the point of view of finite element approximation. As benchmark cases, we analyze in detail three simple cylindrical shell problems. These problems are seemingly similar but turn out to be quite different. We (1) compute the exact displacement and strain fields in the model problems both numerically and symbolically using two variants of classical shell models, (2) for comparison, compute numerically the displacements and strains corresponding to certain higher-order dimension reduction models derived from the energy principle, (3) demonstrate the dissimilarity of the three benchmark problems as to their asymptotic behaviour in the limit of zero thickness and (4) show, both theoretically and experimentally, how this diversity is reflected in the accuracy of standard finite element approximations of both low and high order.

Fourier mode analysis of layers in shallow shell deformations

SummaryWe set up a framework for analyzing mixed finite element methods for the plate problem using a mesh dependent energy norm which applies both to the Kirchhoff and to the Mindlin-Reissner formulation of the problem. The analysis techniques are applied to some low order finite element schemes where three degrees of freedom are associated to each vertex of a triangulation of the domain. The schemes proceed from the Mindlin-Reissner formulation with modified shear energy.