Volume 20, Issue 1

The p -Stokes equations with power-law exponent p ∈ ( 1 , 2 ) {p\in(1,2)} describes non-Newtonian, shear-thinning, incompressible flow. In many industrial applications and natural settings, shear-thinning flow takes place in very thin domains. To account for such anisotropic domains in simulations, we here study an equal-order bi-linear anisotropic finite element discretization of the p -Stokes equations, and extend a non-linear Local Projection Stabilization to anisotropic meshes. We prove an a priori estimate and illustrate the results with two numerical examples, one confirming the rate of convergence predicted by the a-priori analysis, and one showing the advantages of an anisotropic stabilization compared to an isotropic one.

In the present paper, the first and second order of accuracy difference schemes for the approximate solutions of the initial value problem for Schrödinger equation with time delay in a Hilbert space are presented. The theorem on stability estimates for the solutions of these difference schemes is established. The application of theorems on stability of difference schemes for the approximate solutions of the initial boundary value problems for Schrödinger partial differential equation is provided. Additionally, some illustrative numerical results are presented.

We consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, H ⁢ ( div ) {H(\mathrm{div)}} -conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the Arnold–Falk–Winther finite element spaces. We distinguish two stress reconstructions: one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with an analytical solution. We then apply the adaptive algorithm to a more application-oriented test, considering the Hencky–Mises and an isotropic damage model.

We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate | λ k - λ k , h | ≤ C k ⁢ h 2 {\lvert\lambda_{k}-\lambda_{k,h}\rvert\leq C_{k}h^{2}} in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, λ k {\lambda_{k}} and λ k , h {\lambda_{k,h}} are the exact and approximate k th largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that | λ k , h ( G ) - λ k , h ( N ) | ≤ C ⁢ h 2 {|\lambda^{(G)}_{k,h}-\lambda^{(N)}_{k,h}|\leq Ch^{2}} , where λ k , h ( G ) {\lambda^{(G)}_{k,h}} and λ k , h ( N ) {\lambda^{(N)}_{k,h}} denote the k th largest eigenvalues computed by Galerkin and Nyström methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.

The aim of this paper is to provide new perspectives on relative finite element accuracy which is usually based on the asymptotic speed of convergence comparison when the mesh size h goes to zero. Starting from a geometrical reading of the error estimate due to the Bramble–Hilbert lemma, we derive two probability distributions that estimate the relative accuracy, considered as a random variable, between two Lagrange finite elements P k {P_{k}} and P m {P_{m}} ( k < m {k<m} ). We establish mathematical properties of these probabilistic distributions and we get new insights which, among others, show that P k {P_{k}} or P m {P_{m}} is more likely accurate than the other, depending on the value of the mesh size h .

The initial value problem for an evolution equation of type v ′ + A ⁢ v + B ⁢ K ⁢ v = f {v^{\prime}+Av+BKv=f} is studied, where A : V A → V A ′ {A:V_{A}\to V_{A}^{\prime}} is a monotone, coercive operator and where B : V B → V B ′ {B:V_{B}\to V_{B}^{\prime}} induces an inner product. The Banach space V A {V_{A}} is not required to be embedded in V B {V_{B}} or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

In this study, we consider an inverse problem of recovering the initial value for a multi-dimensional time-fractional diffusion-wave equation. By using some additional boundary measured data, the uniqueness of the inverse initial value problem is proven by the Laplace transformation and the analytic continuation technique. The inverse problem is formulated to solve a Tikhonov-type optimization problem by using a finite-dimensional approximation. We test four numerical examples in one-dimensional and two-dimensional cases for verifying the effectiveness of the proposed algorithm.

We perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened 4 / 3 {4/3} -CFL condition ( Δ ⁢ t ∼ h 4 / 3 {\Delta t\sim h^{4/3}} ). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in L 2 {L^{2}} -sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree l ≥ 1 {l\geq 1} under the standard CFL condition.

Image registration is the process of aligning sets of similar, but different, intensity image functions to track changes between the images. In medical image problems involving lung images, variational registration models are a very powerful tool which can aid in effective treatment of various lung conditions and diseases. However, a common drawback of many variational models, such as the diffusion model and even optic flow models, is the lack of control of folding in the deformations leading to physically inaccurate transformations. For this reason, such models are generally not suitable for real life lung imaging problems where folding cannot occur. There are two approaches offering reliable solutions (though not necessarily accurate). The first approach is a parametric model such as the affine registration model, still widely used in many applications, but is unable to track local changes or yield accurate results. The second approach is to impose an extra constraint on the transformation of registration at the cost of increased non-linearity. An alternative to the second approach, achieving diffeomorphic transforms without adding any constraints, is an inverse consistent model such as by Christensen and Johnson (2001) from computing explicitly both the forward and inverse transforms. However, one must deal with the strong non-linearity in the formulation. In this paper we first propose a simplified inverse consistent model to avoid the inclusion of strong non-linearities and then a fast non-linear multigrid (NMG) technique to overcome the extra computational work required by the inverse consistent model. Experiments, performed on real medical CT images, show that our proposed inverse consistent model is robust to both parameter choice and non-folding in the transformations when compared with diffusion type models.

For the Poisson problem in two dimensions, posed on a domain partitioned into axis-aligned rectangles with up to one hanging node per edge, we envision an efficient error reduction step in an instance-optimal hp -adaptive finite element method. Central to this is the problem: Which increase in local polynomial degree ensures p -robust contraction of the error in energy norm? We reduce this problem to a small number of saturation problems on the reference square, and provide strong numerical evidence for their solution.