Carsten Carstensen

Non–convex potentials and microstructures in finite–strain plasticity

A mathematical model for a finite–strain elastoplastic evolution problem is proposed in which one time–step of an implicit time–discretization leads to generally non–convex minimization problems. The elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations. The results for a single slip–system and von Mises plasticity illustrate that finite–strain elastoplasticity generates reduced problems with non–quasiconvex energy densities and so allows for non–attainment of energy minimizers and microstructures.

A posteriori error estimate for the mixed finite element method

A computable error bound for mixed finite element methods is established in the model case of the Poisson-problem to control the error in the H(div,Ω) x L 2 (Ω)-norm. The reliable and efficient a posteriori error estimate applies, e.g., to Raviart-Thomas, Brezzi-Douglas-Marini, and Brezzi-Douglas-Fortin-Marini elements.

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids part I: Low order conforming, nonconforming, and mixed FEM

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

Remarks around 50 lines of Matlab: short finite element implementation

A short Matlab implementation for P1-x1 finite elements on triangles and parallelograms is provided for the numerical solution of elliptic problems with mixed boundary conditions on unstructured grids. According to the shortness of the program and the given documentation, any adaptation from simple model examples to more complex problems can easily be performed. Numerical examples prove the flexibility of the Matlab tool.

Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods

We prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on the error of linear finite element methods both in H1- and L2-norms. We present two proofs: one uses the standard L2-projection and the other relies on a new, weighted Clement-type interpolation operator.

Numerical solution of the scalar double-well problem allowing microstructure

The direct numerical solution of a non-convex variational problem (P) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by numerical solution of the relaxed problem (RP) leading to a (degenerate) convex minimisation problem. The problem (RP) has a minimiser u and a related stress field σ = DW**(⊇u) which is known to coincide with the stress field obtained by solving (P) in a generalised sense involving Young measures. If u h is a finite element solution, σ h := DW**(⊇u h ) is the related discrete stress field. We prove a priori and a posteriori estimates for σ-σ h in L 4/3 (Ω) and weaker weighted estimates for ⊇u - ⊇u h . The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments.

Matlab implementation of the finite element method in elasticity

Abstract A short Matlab implementation for P1 and Q1 finite elements (FE) is provided for the numerical solution of 2d and 3d problems in linear elasticity with mixed boundary conditions. Any adaptation from the simple model examples provided to more complex problems can easily be performed with the given documentation. Numerical examples with postprocessing and error estimation via an averaged stress field illustrate the new Matlab tool and its flexibility.

An a posteriori error estimate for a first-kind integral equation

In this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the first kind. The estimate is local and sharp for quasi-uniform meshes and so improves earlier work of ours. The mesh-dependence of the constants is analyzed and shown to be weaker than expected from our previous work. Besides the Galerkin boundary element method, the collocation method and the qualocation method are considered. A numerical example is given involving an adaptive feedback algorithm.

Averaging technique for FE – a posteriori error control in elasticity. Part I: Conforming FEM

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second-order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, the reliability of any averaging estimator is shown for low order finite element methods in elasticity. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides and independent of the structure of a shape-regular mesh.

Convergence analysis of an adaptive nonconforming finite element method

An adaptive nonconforming finite element method is developed and analyzed that provides an error reduction due to the refinement process and thus guarantees convergence of the nonconforming finite element approximations. The analysis is carried out for the lowest order Crouzeix-Raviart elements and leads to the linear convergence of an appropriate adaptive nonconforming finite element algorithm with respect to the number of refinement levels. Important tools in the convergence proof are a discrete local efficiency and a quasi-orthogonality property. The proof does neither require regularity of the solution nor uses duality arguments. As a consequence on the data control, no particular mesh design has to be monitored.