Volume 22, Issue 2

The gradient discretisation method (GDM) is a generic framework designed recently, as a discretisation in spatial space, to partial differential equations. This paper aims to use the GDM to establish a first general error estimate for numerical approximations of parabolic obstacle problems. This gives the convergence rates of several well-known conforming and non-conforming numerical methods. Numerical experiments based on the hybrid finite volume method are provided to verify the theoretical results.

The paper is concerned with functional-type a posteriori estimates for the initial boundary value problem for a parabolic partial differential equation with an obstacle. We deduce a guaranteed and computable bound of the distance between the exact minimizer and any function from the admissible (energy) class of functions. Applications to the analysis of modeling errors caused by data implification are discussed. An important case of time incremental approximations is specially studied. Numerical examples presented in the last section show how the estimates work in practice.

Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issue, and developing simpler but accurate approximate models is still worthwhile. Here, we propose a new family of paraxial asymptotic models that approximates the Vlasov–Maxwell system of equations. We introduce parameters in our models that allow us to handle relativistic cases, much slower beams or even non-relativistic cases. These models are derived by introducing a small parameter and provide static or quasi-static approximate equations that are 𝑛-th order accurate; 𝑛 may be chosen as required. Practically, one can select a model by determining the regime one is interested in and choosing the degree of accuracy needed.

In this paper, a penalty formulation is proposed and analyzed in both continuous and finite element setups, for the two-dimensional Oldroyd model of order one, when the initial velocity is in 𝐇 0 1 {\mathbf{H}_{0}^{1}} . New regularity results which are valid uniformly in time as t → ∞ {t\to\infty} and in the penalty parameter ε as ε → 0 {\varepsilon\to 0} are derived for the solution of the penalized problem. Then, based on conforming finite elements to discretize the spatial variables and keeping temporal variable continuous, a semidiscrete problem is discussed and a uniform-in-time a priori bound of the discrete velocity in Dirichlet norm is derived with the help of a penalized discrete Stokes operator and a modified uniform Gronwall’s lemma. Further, optimal error estimates for the penalized velocity in 𝐋 2 {\mathbf{L}^{2}} as well in 𝐇 1 {\mathbf{H}^{1}} -norms and for the penalized pressure in L 2 {L^{2}} -norm have been established for the semidiscrete problem with non-smooth data. These error estimates hold uniformly in time under uniqueness assumption and also in the penalty parameter as it goes to zero. Our analysis relies on the suitable use of the inverse of the penalized Stokes operator, penalized Stokes–Volterra projection and judicious application of weighted time estimates with positivity property of the memory term. Finally, several numerical experiments are conducted on benchmark problems which confirm our theoretical findings.

We consider the numerical solution of elliptic problems in 3D with boundary nonlinearity, such as arising in stationary heat conduction models. We allow general non-orthotropic materials where the matrix of heat conductivities is a nondiagonal full matrix. The solution approach involves the finite element method (FEM) and Newton type iterations. We develop a quasi-Newton method for this problem, using spectral equivalence to approximate the derivatives. We derive the convergence of the method, and numerical experiments illustrate the robustness and the reduced computational cost.

We propose and investigate two regularization models for fluid flows at higher Reynolds numbers. Both models are based on the reduced ADM regularization (RADM). One model, which we call DC-RADM (deferred correction for reduced approximate deconvolution model), aims to improve the temporal accuracy of the RADM. The second model, denoted by RADC (reduced approximate deconvolution with correction), is created with a more systemic approach. We treat the RADM regularization as a defect in approximating the true solution of the Navier–Stokes equations (NSE) and then correct for this defect, using the defect correction algorithm. Thus, the resulting RADC model can be viewed as a first member of the class that we call “LESC-reduced”, where one starts with a regularization that resembles a Large Eddy Simulation turbulence model and then improves it with a defect correction technique. Both models are investigated theoretically and numerically, and the RADC is shown to outperform the DC-RADM model both in terms of convergence rates and in terms of the quality of the produced solution.

In this work, we consider a fully discrete finite element approximation of the 3D incompressible magnetohydrodynamic system. The velocity and magnetic field are approximated both by piecewise quadratic finite elements, while the pressure is approximated by piecewise linear finite elements. The time discretization is based on the Crank–Nicolson scheme for the linear terms in the model and the explicit Adams–Bashforth for the nonlinear terms. We establish the optimal error estimates of both the approximate velocity and magnetic field in H 1 \mathbf{H}^{1} -norm and of the approximate pressure in L 2 L^{2} -norm. In order to achieve the optimal L 2 L^{2} -norm error estimates of both the approximate velocity and magnetic field, we shall make use of a special negative norm technique.

This paper is concerned with the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated ℓ p {\ell^{p}} -quasi-norm minimization. Compared to ℓ 1 {\ell^{1}} -norm minimization, the choice of 0 < p < 1 {0<p<1} provides a natural framework to accommodate usual constraints which quadrature rules must fulfil. We also extend an a priori error estimate available for the ℓ 1 {\ell^{1}} -norm formulation by considering the error resulting from data compression. Finally, we present numerical examples to investigate the numerical performance of our method and compare our results to both ℓ 1 {\ell^{1}} -norm minimization and nonnegative least squares method. Matlab codes related to the numerical examples and the algorithms described are provided.

An inverse problem of identifying the diffusion coefficient in matrix form in a parabolic PDE is considered. Following the idea of natural linearization, considered by Cao and Pereverzev (2006), the nonlinear inverse problem is transformed into a problem of solving an operator equation where the operator involved is linear. Solving the linear operator equation turns out to be an ill-posed problem. The method of Tikhonov regularization is employed for obtaining stable approximations and its finite-dimensional analysis is done based on the Galerkin method, for which an orthogonal projection on the space of matrices with entries from L 2 ⁢ ( Ω ) L^{2}(\Omega) is defined. Since the error estimates in Tikhonov regularization method rely heavily on the adjoint operator, an explicit representation of adjoint of the linear operator involved is obtained. For choosing the regularizing parameter, the adaptive technique is employed in order to obtain order optimal rate of convergence. For the relaxed noisy data, we describe a procedure for obtaining a smoothed version so as to obtain the error estimates. Numerical experiments are carried out for a few illustrative examples.

This paper studies a model of two-phase flow with an immersed material viscous interface and a finite element method for the numerical solution of the resulting system of PDEs. The interaction between the bulk and surface media is characterized by no-penetration and slip with friction interface conditions. The system is shown to be dissipative, and a model stationary problem is proved to be well-posed. The finite element method applied in this paper belongs to a family of unfitted discretizations. The performance of the method when model and discretization parameters vary is assessed. Moreover, an iterative procedure based on the splitting of the system into bulk and surface problems is introduced and studied numerically.

We consider fourth-order singularly perturbed eigenvalue problems in one-dimension and the approximation of their solution by the h version of the Finite Element Method (FEM). In particular, we use a C 1 {C^{1}} -conforming FEM with piecewise polynomials of degree p ≥ 3 {p\geq 3} defined on an exponentially graded mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at the optimal rate when the error in the eigenvalues is measured in absolute value and the error in the eigenvectors is measured in the energy norm. We also illustrate our theoretical findings through numerical computations for the case p = 3 {p=3} .

The problem of numerical differentiation for periodic bivariate functions with finite smoothness is studied. To achieve stable approximations, we investigate some variants of the Fourier truncation method. Estimates of the accuracy and volume of the used Fourier coefficients are found for the constructed methods. We perform numerical experiments that confirm correctness of our theoretical conclusions.

Many non-local processes are modeled using mathematical models that include fractional powers of elliptic operators. The approximate solution of stationary problems with fractional powers of operators is most often based on rational approximations introduced in various versions for a fractional power of the self-adjoint positive operator. The purpose of this work is to use such approximations for the approximate solution of nonstationary problems. We consider Cauchy problems for the first and second order differential-operator equations in finite-dimensional Hilbert spaces. Estimates for the proximity of an approximate solution to an exact one are obtained when specifying the absolute and relative errors of the approximation of the fractional power of the operator. We construct splitting schemes based on the additive representation with a rational approximation of the operator’s fractional power. The stability and accuracy of factorized two-level additive operator-difference schemes for the first order evolution equation and three-level schemes for a second order equation are established.