Volume 17, Issue 3

In 2016, the biennial conference Computational Methods in Applied Mathematics (CMAM) was dedicated to a remarkable event: the hundredth anniversary of the Galerkin method. This special volume of the same name journal is mainly based on the papers of participants of this conference. The introductory article contains a brief description of the origin and development of the Galerkin method and gives an overview of the conference, which was held at the University of Jyväskylä (Finland), July 31 – August 6, 2016.

In this work, we introduce and analyze an hp -hybrid high-order ( hp -HHO) method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We prove hp -convergence estimates for both the energy- and L2L^{2}-norms of the error, which are the first of this kind for Hybrid High-Order methods. These results hinge on a novel hp -approximation lemma valid for general polytopal elements in arbitrary space dimension. The estimates are additionally fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on the square root of the local anisotropy, improving previous results for HHO methods. The expected exponential convergence behavior is numerically demonstrated on a variety of meshes for both isotropic and strongly anisotropic diffusion problems.

We discuss the construction of robust preconditioners for finite element approximations of Biot’s consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger–Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.

In this article, two novel numerical methods have been developed for simulating fluid/porous particle interactions in three-dimensional (3D) Stokes flow. The Brinkman–Debye–Bueche model is adopted for the fluid flow inside the porous particle, being coupled with the Stokes equations for the fluid flow outside the particle. The rotating motion of a porous ball and the interaction of two porous balls in bounded shear flows have been studied by these two new methods. The numerical results show that the porous particle permeability has a strong effect on the interaction of two porous balls.

We discuss the differences between the penalty, mixed and stabilised methods for the finite element approximation of the obstacle problem. The theoretical properties of the methods are discussed and illustrated through numerical examples.

This paper introduces and analyzes the new grid-based tensor approach to approximate solutions of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree–Fock equation over a spatial L1×L2×L3{L_{1}\times L_{2}\times L_{3}} lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large L1×L2×L3{L_{1}\times L_{2}\times L_{3}} lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic L×1×1{L\times 1\times 1} lattice chain in a 3D rectangular “tube” with L up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large L .

We consider an iteration method for solving an elliptic type boundary value problem 𝒜⁢u=f{\mathcal{A}u=f}, where a positive definite operator 𝒜{\mathcal{A}} is generated by a quasi-periodic structure with rapidly changing coefficients (a typical period is characterized by a small parameter ϵ). The method is based on using a simpler operator 𝒜0{\mathcal{A}_{0}} (inversion of 𝒜0{\mathcal{A}_{0}} is much simpler than inversion of 𝒜{\mathcal{A}}), which can be viewed as a preconditioner for 𝒜{\mathcal{A}}. We prove contraction of the iteration method and establish explicit estimates of the contraction factor q . Certainly the value of q depends on the difference between 𝒜{\mathcal{A}} and 𝒜0{\mathcal{A}_{0}}. For typical quasi-periodic structures, we establish simple relations that suggest an optimal 𝒜0{\mathcal{A}_{0}} (in a selected set of “simple” structures) and compute the corresponding contraction factor. Further, this allows us to deduce fully computable two-sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient if the coefficients of 𝒜{\mathcal{A}} admit low-rank representations and if algebraic operations are performed in tensor structured formats. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely linear-logarithmically) on the frequency parameter 1ϵ\frac{1}{\epsilon}.

This paper contains an error analysis of two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable. The main result consists of precise bounds for the discretization error with respect to the Lp⁢(Ω;ℝd){L^{p}(\Omega;{\mathbb{R}}^{d})}-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

We consider a coupled system of Maxwell’s equations and the equations of elasticity, which is commonly used to model piezo-electric material behavior. The boundary influence is encoded as a separate dynamics on the boundary data spaces coupled to the partial differential equations. Evolutionary well-posedness, i.e. Hadamard well-posedness and causal dependence on the data, is shown for the resulting model system.

We consider elliptic problems with complicated, discontinuous diffusion tensor A0{A_{0}}. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say Aε{A_{\varepsilon}}, and to use standard finite elements. In [19] a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error was derived under the assumption that the difference A0-Aε{A_{0}-A_{\varepsilon}} becomes small with respect to the L∞{L^{\infty}}-norm. This implies in particular that interfaces/discontinuities separating the smooth parts of A0{A_{0}} have to be matched exactly by the coefficient Aε{A_{\varepsilon}}. Therefore the efficient application of that theory to problems with complicated or curved interfaces is limited. In this paper, we will present a refined theory, where the difference A0-Aε{A_{0}-A_{\varepsilon}} is measured in the Lq{L^{q}}-norm for some appropriate q∈]2,∞[{q\in{]2,\infty[}} and, hence, the geometric resolution condition is significantly relaxed.