The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version of the Picard iteration method. The accuracy of the proposed algorithm is investigated.

In this paper we analyze the effect of introducing a numerical integration in the piecewise linear finite element approximation of the Steklov eigenvalue problem. We obtain optimal order error estimates for the eigenfunctions when this numerical integration is used and we prove that, for singular eigenfunctions, the eigenvalues obtained using this reduced integration are better approximations than those obtained using exact integration when the mesh size is small enough.

Arbitrage-free prices u of European contracts on risky assets whoselog-returns are modelled by Lévy processes satisfya parabolic partial integro-differential equation (PIDE) $\partial_t u + {\mathcal{A}}[u] = 0$ .This PIDE is localized tobounded domains and the error due to this localization isestimated. The localized PIDE is discretized by theθ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for ${\mathcal{A}}$ can be replaced by a sparse matrix in the wavelet basis, and the linear systemsin each implicit time step are solved approximativelywith GMRES in linear complexity. The total work of the algorithm for M time steps is bounded byO(MN(log(N))2) operations and O(Nlog(N)) memory.The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solutionin the same complexity as finite difference approximationsof the standard Black–Scholes equation.Computational examples for various Lévy price processes are presented.

Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces in case of hexahedral triangulations. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We present the discretization errors in different norms for linear and quadratic mortar finite elements with different Lagrange multiplier spaces. Numerical results illustrate the performance of our approach.

We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega\subset \mathbb{R}^d$ and for T > 0 indimension dd ≥ 1. We use a wavelet based sparse gridspace discretization with mesh-width h and order pd ≥ 1, andhp discontinuous Galerkin time-discretization of order $r =O(\left|\log h\right|)$ on a geometric sequence of $O(\left|\log h\right|)$ many timesteps. The linear systems in each time step are solved iterativelyby $O(\left|\log h\right|)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2(Ω)-error of O(N-p) for u(x,T) where N is the total number of operations,provided that the initial data satisfies $u_0 \in H^\varepsilon(\Omega)$ with ε > 0 and that u(x,t) is smooth in x for t>0. Numerical experiments in dimension d up to 25 confirm thetheory.

Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictatesthe interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε-1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.

A Volterra model with mutual interferenceconcerning integrated pest management is proposed and analyzed. Byusing Floquet theorem and small amplitude perturbation method andcomparison theorem, we show the existence of a globallyasymptotically stable pest-eradication periodic solution. Further,we prove that when the stability of pest-eradication periodicsolution is lost, the system is permanent and there exists alocally stable positive periodic solution which arises from thepest-eradication periodic solution by bifurcation theory. When theunique positive periodic solution loses its stability, numericalsimulation shows there is a characteristic sequence ofbifurcations, leading to a chaotic dynamics. Finally, we comparethe validity of integrated pest management (IPM) strategy withclassical methods and conclude IPM strategy is more effective thanclassical methods.

We present an integral equation method for solving boundary valueproblems of the Helmholtz equation in unbounded domains. Themethod relies on the factorisation of one of the Calderón projectors by an operator approximating the exterioradmittance (Dirichlet to Neumann) operator of the scatteringobstacle. We show how the pseudo-differential calculus allows usto construct such approximations and that this yields integralequations without internal resonances and being well-conditionedat all frequencies. An implementation technique is elaborated,where again reasonings from pseudo-differential calculus play animportant rôle. Some numerical examples are presented which appearto confirm that the new integral equation leads to linear systemswhich are much better conditioned than the classical ("direct")integral equations and hence have much better behaviour whensolved with iterative techniques and matrix sparsification.

A general setting is proposed for the mixed finite element approximations of elliptic differential problems involving a unilateral boundary condition. The treatment covers the Signorini problem as well as the unilateral contact problem with or without friction. Existence, uniqueness for both the continuous and the discrete problem as well as error estimates are established in a general framework. As an application, the approximation of the Signorini problem by the lowest order mixed finite element method of Raviart–Thomas is proved to converge with a quasi-optimal error bound.