Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem

In this paper, a least squares group finite element method for solving coupled Burgers' problem in 2-D is presented. A fully discrete formulation of least squares finite element method is analyzed, the backward-Euler scheme for the time variable is considered, the discretization with respect to space variable is applied as biquadratic quadrangular elements with nine nodes for each element. The continuity, ellipticity, stability condition and error estimate of least squares group finite element method are proved. The theoretical results show that the error estimate of this method is . The numerical results are compared with the exact solution and other available literature when the convection-dominated case to illustrate the efficiency of the proposed method that are solved through implementation in MATLAB R2018a.

A layer potential approach to functional and clinical brain imaging

In this work, we consider the inverse source recovery problem from sEEG, EEG and MEG point-wise data. We regard this as an inverse source recovery problem for L2 vector-fields normally oriented and supported on the grey/white matter interface, which together with the brain, skull and scalp form a non-homogeneous layered conductor. We assume that the quasistatic approximation of Maxwell’s equation holds for the electro-magnetic fields considered. The electric data is measured point-wise inside and outside the conductor while the magnetic data is measured only point-wise outside the conductor. These ill-posed problems are solved via Tikhonov regularization on triangulations of the interfaces and a piecewise linear model for the current on the triangles. Both in the continuous and discrete formulation the electric potential is expressed as a linear combination of double layer potentials while the magnetic flux density in the continuous case is a vector-surface integral whose discrete formulation features single layer potentials. A main feature of our approach is that these contributions can be computed exactly. Due to the consideration of the regularity conditions of the electric potential in the inverse source recovery problem, the Cauchy transmission problem for the electric potential is inadvertently solved as well. In the problem, we propagate only the electric potential while the normal derivatives at the interfaces of discontinuity of the electric conductivities are computed directly from the resulting solution. This reduces the computational complexity of the problem. There is a direct connection between the magnetic flux density and the electrical potential in conductors such as the one we explore, hence a coupling of the sEEG, EEG and MEG data for solving the respective inverse source recovery problems simultaneously is direct. We treat these problems in a unified approach that uses only single and/or double layer potentials. We provide numerical examples using realistic meshes of the head with synthetic data.

The curl–curl conforming virtual element method for the quad-curl problem

In this paper, we present the [Formula: see text]-conforming virtual element (VE) method for the quad-curl problem in two dimensions. Based on the idea of de Rham complex, we first construct three families of [Formula: see text]-conforming VEs, of which the simplest one has only one degree of freedom associated to each vertex and each edge in the lowest-order case, respectively. An exact discrete complex is established between the [Formula: see text]-conforming and [Formula: see text]-conforming VEs. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the coercivity and inf–sup condition of the corresponding discrete formulation. We show that the conforming VEs have the optimal convergence. Some numerical examples are given to confirm the theoretical results.

Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

This work deals with the efficient iterative solution of the system of equations stemming from mimetic finite difference discretization of a hybrid-dimensional mixed Darcy problem modeling flow in fractured porous media. We investigate the spectral properties of a mixed discrete formulation based on mimetic finite differences for flow in the bulk matrix and finite volumes for the fractures, and present an approximation of the factors in a set of approximate block factorization preconditioners that accelerates convergence of iterative solvers applied to the resulting discrete system. Numerical tests on significant three-dimensional cases have assessed the properties of the proposed preconditioners.

Time-Limited Codewords over Band-Limited Channels: Data Rates and the Dimension of the W-T Space

We consider a communication system whereby T-seconds time-limited codewords are transmitted over a W-Hz band-limited additive white Gaussian noise channel. In the asymptotic regime as WT→∞, it is known that the maximal achievable rates with such a scheme converge to Shannon’s capacity with the presence of 2WT degrees of freedom. In this work we study the degrees of freedom and the achievable information rates for finite values of WT. We use prolate spheroidal wave functions to obtain an information lossless equivalent discrete formulation and then we apply Polyanskiy’s results on coding in the finite block-length regime. We derive upper and lower bounds on the achievable rates and the corresponding degrees of freedom and we numerically evaluate them for sample values of 2WT. The bounds are asymptotically tight and numerical computations show the gap between them decreases as 2WT increases. Additionally, the possible decrease from 2WT in the available degrees of freedom is upper-bounded by a logarithmic function of 2WT.

An Immersed Finite Element Method for Planar Elasticity Interface Problem

This paper presents the P1/CR immersed finite element (IFE) method to solve planar elasticityinterface problem. By adding some stabilisation terms on the edges of interface elements, thestability of the discrete formulation and a priori error estimate in an energy norm are presented.Finally, numerical examples are given to confirm our theoretical results.

APPLICATION OF THE DEDUCTIONS FROM NAVIER STOKES EQUATIONS FOR THE DETERMINATION OF FLOW VELOCITY AND THROUGHPUT IN A GAS PIPELINE BY COMPUTATIONAL APPROACH

Theoretical treatment of gas pipeline pressure-flow problem had been presented applying Navier Stokes equation reduced to their appropriate forms by applicable practical conditions. The results obtained from the theoretical analysis tally with the operating conditions of the case study pipelines. The pipelines being Shell Petroleum Development Company (SPDC) and ElfTotal Nigeria Limited. The results obtained by numerical discretization suggested that these pipelines are not optimally operated. Hence, the need to adjust the flow situation to bring pressure and flow throughput to optimal level of performance. Throughput in excess of the operating conditions could be accommodated by these operating pipelines. It is imperative that this could prevent the spread of these vital capital intensive assets. The funds so conserved could be diverted to sourcing for new gas fields to increase the nation’s strategic reserves.Purpose: The purpose of this work is to enable comparative analysis of the results of the deductions from Nervier Stokes equations with that generated by computer simulation of the discrete formulation.Methodology: Outlining the deductions and developing the discrete formulation. Computer program was developed for the discrete formulation and operational data from operating gas pipelines injected both for the deductions and computational algorithmic coding and the deduced expressions from the Nervier Stokes equations. Results obtained were compared in a bid to address line throughput subject to the operational conditions of the specified gas pipelines in this study.Findings: The output results of the Nervier Stokes deductions matched closed with operational throughput of the two gas pipelines. The numerical discretization simulation results confirmed that additional throughput far and above 1.8m3/s could still be accommodated by these gas pipelines.Unique contribution to theory, practice and policy: As earlier predicted, our existing gas pipelines are grossly under-operated. Additional capacity much more than the operational capacity could still be accommodated by these gas pipelines.