Band 22, Heft 3

In this paper we provide some more details on the numerical analysis and we present some enlightening numerical results related to the spectrum of a finite element least-squares approximation of the linear elasticity formulation introduced recently. We show that, although the formulation is robust in the incompressible limit for the source problem, its spectrum is strongly dependent on the Lamé parameters and on the underlying mesh.

The elliptic problem - div ⁡ ( μ ⁢ ∇ ⁡ u ) = f -\operatorname{div}(\mu\nabla u)=f is considered, where μ > 0 \mu>0 is smooth but strongly varying. Anisotropic a posteriori error estimates are derived, the effectivity index being bounded above and below by two constants independent of the data 𝑓, 𝜇, the mesh size and aspect ratio, up to higher order terms. Numerical experiments on non-adapted and adapted anisotropic meshes confirm these predictions.

We study a fourth-order div problem and its approximation by the discontinuous Petrov–Galerkin method with optimal test functions. We present two variants, based on first and second-order systems. In both cases, we prove well-posedness of the formulation and quasi-optimal convergence of the approximation. Our analysis includes the fully-discrete schemes with approximated test functions, for general dimension and polynomial degree in the first-order case, and for two dimensions and lowest-order approximation in the second-order case. Numerical results illustrate the performance for quasi-uniform and adaptively refined meshes.

We report on the Matlab program package IGABEM2D which provides an easily accessible implementation of adaptive Galerkin boundary element methods in the frame of isogeometric analysis and which is available on the web for free download. Numerical experiments with IGABEM2D underline the particular importance of adaptive mesh refinement for high accuracy in isogeometric analysis.

An error estimate is presented for the Newton iterative Crank–Nicolson finite element method for the nonlinear Schrödinger equation, fully discretized by quadrature, without restriction on the grid ratio between temporal step size and spatial mesh size. It is shown that the Newton iterative solution converges double exponentially with respect to the number of iterations to the solution of the implicit Crank–Nicolson method uniformly for all time levels, with optimal convergence in both space and time.

We investigate the discretization of H ⁢ ( curl ) H(\mathrm{curl}) and H ⁢ ( div ) H(\mathrm{div}) in two and three space dimensions by partially discontinuous nodal finite elements, i.e., vector-valued Lagrange finite elements with discontinuity in certain directions. These spaces can be implemented as a combination of continuous and discontinuous Lagrange elements and fit in de Rham complexes. We construct well-conditioned nodal bases.

In this study, we present new sharper constructive a priori error estimates for a full-discrete numerical solution of the heat equation that refines our previous work. The full discretization is given using the finite element method in space and linear interpolation in time, similar to that in the previous work. In particular, we adopt an approach based on the effective use of the properties both for exact and discretized time-periodic solutions to establish the error estimates simpler and sharper than the previous work. Furthermore, we derive some time-stepwise error estimates in the space direction. Finally, we present several numerical examples that confirm the actual refinements of the convergence.

This article summarizes the L p L^{p} -DPG method presented in [18], where only 1D convection-diffusion problems are solved. We apply the same computational techniques to 2D convection-diffusion problems and report additional numerical results herein. Furthermore, we propose an L p L^{p} -DPG method with variable 𝑝 and illustrate it with numerical experiments.

We consider a class of iterative methods based on block splitting (BBS) to solve absolute value equations A ⁢ x - | x | = b Ax-\lvert x\rvert=b . Recently, several works were devoted to deriving sufficient conditions for the convergence of iterative methods of this type under certain assumptions including ν := ∥ A - 1 ∥ < 1 \nu:=\lVert A^{-1}\rVert<1 . However, the BBS-type iterative methods tend to converge slowly when 𝜈 is very close to one (i.e., ν ≈ 1 \nu\approx 1 ). In this paper, using an auxiliary matrix, we develop a new approach by first rewriting the main problem into a new equivalent block system having shifted ( 1 , 1 ) (1,1) -block and then constructing a fixed point iteration. The exploited strategy can significantly improve the convergence speed of the BBS-type iterative methods when ν ≈ 1 \nu\approx 1 . Numerical experiments are reported to demonstrate the superiority of the new modified iterative scheme over the existing original form of BBS-type methods in the literature.

In this paper, we introduce a first-order low-regularity integrator for the Davey–Stewartson system in the elliptic-elliptic case. It only requires the boundedness of one additional derivative of the solution to be first-order convergent. By rigorous error analysis, we show that the scheme provides first-order accuracy in H γ ⁢ ( T d ) H^{\gamma}(\mathbb{T}^{d}) for rough initial data in H γ + 1 ⁢ ( T d ) H^{\gamma+1}(\mathbb{T}^{d}) with γ > d 2 \gamma>\frac{d}{2} .

In this paper, a stochastic age-structured HIV/AIDS model with nonlinear incidence rates is proposed. It is of great importance to develop efficient numerical approximation methods to solve this HIV/AIDS model since most stochastic partial differential equations (SPDEs) cannot be solved analytically. From the perspective of biological significance, the exact solution of the HIV/AIDS model must be nonnegative and bounded. Then a modified explicit Euler–Maruyama (EM) scheme is constructed based on a projection operator. The EM scheme could preserves the nonnegativity of the numerical solutions and also make the numerical solutions not outside the domain of the exact solutions. The convergence results between the numerical solutions and the exact solutions are analyzed, and some numerical examples are given to verify our theoretical results.

In this article, a space-dependent epidemic model equipped with a constant latency period is examined. We construct a delay partial integro-differential equation and show that its solution possesses some biologically reasonable features. We propose some numerical schemes and show that, by choosing the time step to be sufficiently small, the schemes preserve the qualitative properties of the original continuous model. Finally, some numerical experiments are presented that confirm the aforementioned theoretical results.

In this paper, we present a priori and a posteriori analysis of a staggered discontinuous Galerkin (DG) method for quasi-linear second order elliptic problems of nonmonotone type. First, existence is proved by using Brouwer’s fixed point argument and uniqueness is verified utilizing Lipschitz continuity of the discrete solution map. Next, optimal a priori error estimates for both potential and flux variables are derived. Then the residual based a posteriori error estimates on the potential energy error and the flux L 2 {L^{2}} error, respectively, are proposed. The flux error estimator makes use of a Helmholtz-type decomposition for the nonlinear system, which relies on appropriate choice of an auxiliary problem. While a priori error analysis is based on the observation that the staggered DG method can be viewed as a nonconforming approximation of the primal mixed formulation of the problem, a posteriori error estimation takes advantage of the primal formulation which can be obtained from the primal mixed formulation by eliminating the flux variable in the continuous setting. Finally, the theoretical findings are illustrated by numerical experiments.