Lower-order equivalent nonstandard finite element methods for biharmonic plates

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C0 interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side and then are quasi-optimal in their respective discrete norms. The smoother JI M is defined for a piecewise smooth input function by a (generalized) Morley interpolation I M followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj: Comparison results of nonstandard P 2 finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends and unifies the work [Veeser, Zanotti: Quasioptimal nonconforming methods for symmetric elliptic problems, SIAM J. Numer. Anal. 56 (2018)] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations

In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).

A discrete polynomials approach for optimal control of fractional Volterra integro-differential equations

This study develops an efficient numerical method for optimal control problems governed by fractional Volterra integro-differential equations. A new type of polynomials orthogonal with respect to a discrete norm, namely discrete Hahn polynomials, is introduced and its properties investigated. Fractional operational matrices for the orthogonal polynomials are also derived. A direct numerical algorithm supported by the discrete Hahn polynomials and operational matrices is used to approximate solution of optimal control problems governed by fractional Volterra integro-differential equations. Several examples are analyzed and the results compared with those of other methods. The required CPU time assesses the computational cost and complexity of the proposed method.

Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems

In this work, we prove optimal [Formula: see text]-approximation estimates (with [Formula: see text]) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont–Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an [Formula: see text]-boundedness result for [Formula: see text]-orthogonal projectors on polynomial subspaces. The [Formula: see text]-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these [Formula: see text]-estimates to derive novel error estimates for a Hybrid High-Order (HHO) discretisation of Leray–Lions elliptic problems whose weak formulation is classically set in [Formula: see text] for some [Formula: see text]. This kind of problems appears, e.g. in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by [Formula: see text] the meshsize, we prove that the approximation error measured in a [Formula: see text]-like discrete norm scales as [Formula: see text] when [Formula: see text] and as [Formula: see text] when [Formula: see text].

Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

A Characteristic Finite Difference Domain Decomposition Substructuring Method for Immiscible Displacement Problems in Three-Dimensional Porous Media

Two-phase immiscible displacement in porous media is described by a coupled nonlinear system of an elliptic equation (for the pressure) and a parabolic equation (for the saturation). For the saturation changes much rapidly than the pressure, a more accurate solution (in both time and space) should be illustrated in practical numerica simulaiton for the former unknown. In this paper we present a seven-point central finite difference scheme to simulate the pressure and a characteristic finite difference combinng with domain decomposition method for the saturation equation. This method consists of reduced two-dimensional computation on the subdomain interface boundaries and fully implicit computation parallelly in subdomains. Aparallel algorithm is outlined and an error estimate in discrete norm is derived by introducing new inner products and norms. At the end of this paper, numerical experiments are presented in order to demonstrate theoretical results and the efficiency.