Band 25, Heft 4

For a dual-dual formulation of a frictional contact problem in linear elasticity we present a mixed finite element method based on PEERS elements and we analyze the performance of a nested Uzawa algorithm.

We introduce a new hp -adaptive strategy for self-adjoint elliptic boundary value problems that does not rely on using classical a posteriori error estimators. Instead, our approach is based on a generally applicable prediction strategy for the reduction of the energy error that can be expressed in terms of local modifications of the degrees of freedom in the underlying discrete approximation space. The computations related to the proposed prediction strategy involve low-dimensional linear problems that are computationally inexpensive and highly parallelizable. The mathematical building blocks for this new concept are first developed on an abstract Hilbert space level, before they are employed within the specific context of hp -type finite element discretizations. For this particular framework, we discuss an explicit construction of p -enrichments and hp -refinements by means of an appropriate constraint coefficient technique that can be employed in any dimensions. The applicability and effectiveness of the resulting hp -adaptive strategy is illustrated with some 1- and 2-dimensional numerical examples.

The propagation of charged particles through a scattering medium in the presence of a magnetic field can be described by a Fokker–Planck equation with Lorentz force. This model is studied both from a theoretical and a numerical point of view. A particular trace estimate is derived for the relevant function spaces to clarify the meaning of boundary values. Existence of a weak solution is then proven by the Rothe method. In the second step of our investigations, a fully practical discretization scheme is proposed based on an implicit Euler method for the energy variable and a spherical-harmonics finite-element discretization with respect to the remaining variables. A complete error analysis of the resulting scheme is given and numerical tests are presented to illustrate the theoretical results and the performance of the proposed method.

We develop a finite element method for an elliptic Maxwell boundary value problem on polyhedral domains in ℝ 3 {\mathbb{R}^{3}} with a general topology. Our method is based on a Hodge decomposition approach that leads to standard scalar elliptic problems and elliptic saddle point problems for vector potentials that have previously been investigated in the study of fluid flow problems. We carry out an error analysis that does not involve assumed regularity of the solution and present corroborating numerical results.

We develop a convergence analysis for the simplest finite element method for a model linear-quadratic elliptic distributed optimal control problem with pointwise control and state constraints under minimal assumptions on the constraint functions. We then derive the generalized Karush–Kuhn–Tucker conditions for the solution of the optimal control problem from the convergence results of the finite element method and the Karush–Kuhn–Tucker conditions for the solutions of the discrete problems.

A new method is proposed to provide guaranteed lower bounds for eigenvalues of general second order elliptic operators in any dimension. This method employs a novel generalized Crouzeix–Raviart element which is proved to yield asymptotic lower bounds for eigenvalues of general second order elliptic operators, and a simple post-processing method. As a byproduct, a simple and cheap method is also proposed to obtain guaranteed upper bounds for eigenvalues, which is based on generalized Crouzeix–Raviart element approximate eigenfunctions, an averaging interpolation from the generalized Crouzeix–Raviart element space to the conforming linear element space, and an usual Rayleigh–Ritz procedure. The ingredients for the analysis consist of a crucial projection property of the canonical interpolation operator of the generalized Crouzeix–Raviart element, explicitly computable constants for two interpolation operators. Numerical experiments demonstrate that the guaranteed lower bounds for eigenvalues in this paper are superior to those obtained by the Crouzeix–Raviart element.

In this article, an inverse problem for identifying the initial value of time-fractional telegraph equation is addressed. This is a typical ill-posed problem, meaning the solution does not depend continuously on the data. Besides, we give the conditional stability result based on an a priori bound condition. We utilize the modified quasi-boundary regularization method and the Landweber iterative regularization method to obtain the corresponding approximate solutions, respectively. The convergent error estimates under the a priori regularization parameter selection rule and the a posteriori regularization parameter selection rule are given. Some numerical experiments testify the effectiveness and precision of the proposed methods.

In this paper, we consider simplified iterated Lavrentiev regularization in Hilbert scales for obtaining stable approximation solution of an ill-posed nonlinear equation of the form F ⁢ ( x ) = y {F(x)=y} , where F : 𝒟 ⁢ ( F ) ⊆ X → X {F:\mathcal{D}(F)\subseteq X\to X} is a nonlinear operator on Hilbert space X . We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearly conditions on operator F , we prove convergence of method and obtain a rate of convergence result.

This paper discusses lowest-order nonstandard finite element methods for space discretization and explicit and implicit schemes for time discretization of the biharmonic wave equation with clamped boundary conditions. A modified Ritz projection operator defined on H 0 2 ⁢ ( Ω ) {H^{2}_{0}(\Omega)} ensures error estimates under appropriate regularity assumptions on the solution. Stability results and error estimates of optimal order are established in suitable norms for the semidiscrete and explicit/implicit fully-discrete versions of the proposed schemes. Finally, we report on numerical experiments using explicit and implicit schemes for time discretization and Morley, discontinuous Galerkin, and C 0 {C^{0}} interior penalty schemes for space discretization, that validate the theoretical error estimates.

A potential-robust WG (weak Galerkin) method is introduced for the Maxwell equations. We obtain potential/magnetic-permeability independent error estimates. Optimal-order convergence rates are proved in both the energy norm and the L 2 {L^{2}} norm. Numerical examples verify the theory.

The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from [C. Lehrenfeld and M. Olshanskii, An Eulerian finite element method for PDEs in time-dependent domains, ESAIM Math. Model. Numer. Anal. 53 2019, 2, 585–614] of a solution extension to realise the Eulerian time-stepping scheme. However, a reformulation of the partial differential equation is suggested to derive a scheme which conserves the quantity under consideration exactly on the discrete level. For the spatial discretisation, the paper considers an unfitted finite element method. Ghost-penalty stabilisation is used to realise the discrete solution extension and gives a scheme robust against arbitrary intersections between the mesh and geometry interface. The stability is analysed for both first- and second-order backward differentiation formula versions of the scheme. Several numerical examples in two and three spatial dimensions are included to illustrate the potential of this method.

In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate for the L 2 {L^{2}} -norm of the space-time discretization of SHE by a semi-implicit Euler scheme with respect to time and a TPFA scheme with respect to space and the variational solution of SHE. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.

This paper aims to give a unified construction framework of meshless structure-preserving algorithms to solve the d -dimensional ( d = 1 {d=1} or 2) nonlinear Schrödinger equation. Based on the method of lines, we first derive a finite-dimensional Hamiltonian system by using the radial basis function method of the quasi-interpolation and the technique of left-multiplying a diagonal matrix to discretize the space direction. Then suitable geometric numerical integrations can be used to discretize the time direction, which yields a class of meshless structure-preserving algorithms. In addition to the construction, the structure-preserving properties and their proofs are also provided in detail. Besides the uniform and nonuniform grids, the numerical experiments on the random grids are also emphasized to verify the theoretical research well, which is of great significance for scattering points based on the characteristics of actual problems.