Peter Hansbo

Introduction to Adaptive Methods for Differential Equations

Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719). When, severa ...

An unfitted finite element method, based on Nitsche's method, for elliptic interface problems

In this paper we propose a method for the finite element solution of elliptic interface problem, using an approach due to Nitsche. The method allows for discontinuities, internal to the elements, in the approximation across the interface. We show that optimal order of convergence holds without restrictions on the location of the interface relative to the mesh. Further, we derive a posteriori error estimates for the purpose of controlling functionals of the error and present some numerical examples.

Adaptive finite element methods in computational mechanics

We present a general approach to adaptivity for finite element methods and give applications to linear elasticity, non-linear elasto-plasticity and nonlinear conservation laws, including numerical results.

Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method

We propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes. We show optimal error estimates that are uniform with respect to Poissons ratio. The method is thus locking free. We also introduce an equivalent mixed formulation, allowing for completely incompressible elasticity problems. Numerical results are presented.

Energy norm a posteriori error estimation for discontinuous Galerkin methods

In this paper we present a residual-based a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous Galerkin approximations of elliptic problems. The theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed. We also present some illustrating numerical examples.

Continuous Interior Penalty Finite Element Method for Oseen's Equations

In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207-216] to Oseens equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L2-control of the jump of the gradient over element faces (edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energy-type a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.

The characteristic streamline diffusion method for the time-dependent incompressible Navier-Stokes equations

This paper presents a streamline diffusion finite element method for time-dependent flow problems, with or without free surface, governed by the incompressible Navier-Stokes equations. The method is based on space-time elements, discontinuous in time and continuous in space, which yields a general setting: if the elements are oriented along the characteristic direction in space-time a Lagrangian method is obtained, while if they are fixed the method is Eulerian. Thus the method may be implemented as an arbitrary Lagrangian-Eulerian method, retaining the advantages of the streamline diffusion method on fixed grids. In particular, our method is stable in the whole range of Reynolds numbers and yields the possibility of equal order interpolation for velocity and pressure. Furthermore, since the solution is allowed to be discontinuous in time at discrete time levels, large deformations of the original domain are easily handled, e.g. with remeshing. Numerical results for some 2D-problems are given.

A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes

Summary.In this paper we propose a Lagrange multiplier method for the finite element solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poisson’s equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

The characteristic streamline diffusion method for convection-diffusion problems

The paper describes an approximately characteristic finite element method for the solution of the time-dependent linear scalar convection-diffusion equation. The method is based on space-time elements approximately aligned with the characteristics in space-time. Attention is focused on implementation aspects: avoiding mesh tangling, efficient solution procedures and interpolation. Numerical results for some two-dimensional problems are given.

Adaptive streamline diffusion methods for compressible flow using conservation variables

We consider the streamline diffusion finite element method applied to compressible flow using conservation variables. We propose some adaptive algorithms and present related numerical results.