Quasi-interpolation and a posteriori error analysis in finite element methods
Quasi-Interpolation and A Posteriori Error Analysis in Finite Element Methods
One of the main tools in the proof of residual-based error estimates is a quasi-interpolation operator due to Clément. We modify this operator in the setting of a partition of unity with the effect that the approximation error has a local average zero. This results in a new residual-based error estimate with a volume contribution which is smaller than in the standard estimate. For an elliptic model problem, we discuss applications to conforming, nonconforming and mixed finite element methods. ...
Numerical analysis of a relaxed variational model of hysteresis in two-phase solids
This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with the finite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress field within one time-step. A posteriori error estimates motivate an adaptive mesh-refining algorithm for efficient...
Numerical Analysis of a Relaxed Variational Model of Hysteresis in Two-Phase Solids
This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke The new model itself suggests an implicit time-discretization which is combined with the finite element method in space. error estimates are established for the quasioptimal spatial approximation of the stress field within one time-step. error estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed...
Young-measure approximations for elastodynamics with non-monotone stress-strain relations
Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density . Their time-evolution leads to a nonlinear wave equation with the non-monotone stress-strain relation plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very...
Adaptive coupling of boundary elements and finite elements
Young-Measure approximations for elastodynamics with non-monotone stress-strain relations
Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density . Their time-evolution leads to a nonlinear wave equation with the non-monotone stress-strain relation plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding...
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