Etienne Emmrich

Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity

Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modeled by an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects. Interpreting this IDE as an evolutionary equation of second order, well-posedness in L ∞ (ℝ) as well as jump relations are proved. Moreover, the construction of the micromodulus function from the dispersion relation is studied. A numerical approximation based upon quadrature is suggested and carried out for two examples, one involving jump discontinuities in the initial data corresponding to a Riemann-like problem.

The peridynamic equation and its spatial discretisation

Abstract Different spatial discretisation methods for solving the peridynamic equation of motion are suggested. The methods proposed are tested for a linear microelastic material of infinite length in one spatial dimension. Moreover, the conservation of energy is studied for the continuous as well as discretised problem.

Stability and error of the variable two-step BDF for semilinear parabolic problems

The temporal discretisation of a moderate semilinear parabolic problem in an abstract setting by the two-step backward differentiation formula with variable step sizes is analysed. Stability as well as optimal smooth data error estimates are derived if the ratios of adjacent step sizes are bounded from above by 1.91.

Peridynamics: A Nonlocal Continuum Theory

The peridynamic theory is a nonlocal theory of continuum mechanics based on an integro-differential equation without spatial derivatives, which can be easily applied in the vicinity of cracks, where discontinuities in the displacement field occur. In this paper we give a survey on important analytical and numerical results and applications of the peridynamic theory.

Two-step Bdf Time Discretisation of Nonlinear Evolution Problems Governed by Monotone Operators with Strongly Continuous Perturbations

Abstract The time discretisation of the initial-value problem for a first-order evolution equation by the two-step backward differentiation formula (BDF) on a uniform grid is analysed. The evolution equation is governed by a time-dependent monotone operator that might be perturbed by a time-dependent strongly continuous operator. Well-posedness of the numerical scheme, a priori estimates, convergence of a piecewise polynomial prolongation, stability as well as smooth-data error estimates are provided relying essentially on an algebraic relation that implies the G-stability of the two-step BDF with constant time steps.

The peridynamic equation of motion in non-local elasticity theory

During the last few years, non-local theories in solid mechanics that account for effects of long-range interactions have become topical again. One of these theories is the so-called peridynamic modelling, introduced by Silling [1].

Convergence of a Time Discretisation for Doubly Nonlinear Evolution Equations of Second Order

The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations of second order. This also proves existence of a weak solution. The operator acting on the zero-order term is assumed to be the sum of a linear, bounded, symmetric, strongly positive operator and a nonlinear operator that fulfils a certain growth and a Hölder-type continuity condition. The operator acting on the first-order time derivative is a nonlinear hemicontinuous operator that fulfils a certain growth condition and is (up to some shift) monotone and coercive.

STIFFLY ACCURATE RUNGE{KUTTA METHODS FOR NONLINEAR EVOLUTION PROBLEMS GOVERNED BY A MONOTONE OPERATOR

Stiy accurate implicit Runge{Kutta methods are studied for the time discretisation of nonlinear rst-order evolution equations. The equation is supposed to be governed by a time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fullls a certain growth condi- tion. It is proven that the piecewise constant as well as the piecewise linear interpolant of the time-discrete solution converge towards the exact weak solu- tion, provided the Runge{Kutta method is consistent and satises a stability criterion that implies algebraic stability; examples are the Radau IIA and Lo- batto IIIC methods. The convergence analysis is also extended to problems involving a strongly continuous perturbation of the monotone main part.

Operator Differential-Algebraic Equations Arising in Fluid Dynamics

Abstract. Existence and uniqueness of generalized solutions to initial value problems for a class of abstract differential-algebraic equations (DAEs) is shown. The class of equations covers, in particular, the Stokes and Oseen problem describing the motion of an incompressible or nearly incompressible Newtonian fluid but also their spatial semi-discretization. The equations are governed by a block operator matrix with entries that fulfill suitable inf-sup conditions. The problem data are required to satisfy appropriate consistency conditions. The results in infinite dimensions are compared in detail with those known for the DAEs that arise after semi-discretization in space. Explicit solution formulas are derived in both cases.

Stability and Convergence of the Two-step BDF for the Incompressible Navier-Stokes Problem

The incompressible Navier-Stokes problem is discretised in time by means of the two-step backward differentiation formula with constant step sizEs. Existence and stability ofa time discrete solution are proved as well as the conve.gence of a piecewise polynomial prolongation towards a weak solution. The results presented cover both the twoand three-dimensional cäse. Furthermore, a linearisation that is based upon a modification ofthe convective term using a second-order extrapolation is considered.