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### 3D monolithic finite element approach for aero-thermics processes in industrial furnaces⋆

ESAIM: Proceedings

We consider in this paper a mathematical and numerical model to design an industrial software solution able to handle real complex furnaces configurations in terms of geometries, atmospheres, parts positioning, heat generators and physical thermal phenomena. A three dimensional algorithm based on stabilized finite element methods (SFEM) for solving the momentum, energy, turbulence and radiation equations is presented. An immersed volume method (IVM) for thermal coupling of fluids and solids is introduced...

### A BDDC algorithm for a mixed formulation of flow in porous media.

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

### A BDDC algorithm for flow in porous media with a hybrid finite element discretization.

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

### A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations

Open Mathematics

We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both...

### A defect-correction mixed finite element method for stationary conduction-convection problems.

Mathematical Problems in Engineering

### A fictitious domain method for the numerical two-dimensional simulation of potential flows past sails

ESAIM: Mathematical Modelling and Numerical Analysis

This paper deals with the mathematical and numerical analysis of a simplified two-dimensional model for the interaction between the wind and a sail. The wind is modeled as a steady irrotational plane flow past the sail, satisfying the Kutta-Joukowski condition. This condition guarantees that the flow is not singular at the trailing edge of the sail. Although for the present analysis the position of the sail is taken as data, the final aim of this research is to develop tools to compute the sail...

### A fictitious domain method for the numerical two-dimensional simulation of potential flows past sails

ESAIM: Mathematical Modelling and Numerical Analysis

This paper deals with the mathematical and numerical analysis of a simplified two-dimensional model for the interaction between the wind and a sail. The wind is modeled as a steady irrotational plane flow past the sail, satisfying the Kutta-Joukowski condition. This condition guarantees that the flow is not singular at the trailing edge of the sail. Although for the present analysis the position of the sail is taken as data, the final aim of this research is to develop tools to compute the sail...

### A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme

ESAIM: Mathematical Modelling and Numerical Analysis

We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH...

### A High-Order Unifying Discontinuous Formulation for the Navier-Stokes Equations on 3D Mixed Grids

Mathematical Modelling of Natural Phenomena

The newly developed unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for 3D mixed grids. In the current development, tetrahedrons and triangular prisms are considered. The CPR method can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a more efficient differential form....

### A high-precision algorithm for axisymmetric flow.

Mathematical Problems in Engineering

### A modified Cayley transform for the discretized Navier-Stokes equations

Applications of Mathematics

This paper is concerned with the problem of computing a small number of eigenvalues of large sparse generalized eigenvalue problems. The matrices arise from mixed finite element discretizations of time dependent equations modelling viscous incompressible flow. The eigenvalues of importance are those with smallest real part and are used to determine the linearized stability of steady states, and could be used in a scheme to detect Hopf bifurcations. We introduce a modified Cayley transform of the...

### A Multidimensional fluctuation splitting scheme for the three dimensional Euler equations

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

### A multidimensional fluctuation splitting scheme for the three dimensional Euler equations

ESAIM: Mathematical Modelling and Numerical Analysis

The fluctuation splitting schemes were introduced by Roe in the beginning of the 80's and have been then developed since then, essentially thanks to Deconinck. In this paper, the fluctuation splitting schemes formalism is recalled. Then, the hyperbolic/elliptic decomposition of the three dimensional Euler equations is presented. This decomposition leads to an acoustic subsystem and two scalar advection equations, one of them being the entropy advection. Thanks to this decomposition, the two scalar...

### A network programming approach in solving Darcy's equations by mixed finite-element methods.

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

### A new quadrilateral MINI-element for Stokes equations

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce a new stable MINI-element pair for incompressible Stokes equations on quadrilateral meshes, which uses the smallest number of bubbles for the velocity. The pressure is discretized with the P1-midpoint-edge-continuous elements and each component of the velocity field is done with the standard Q1-conforming elements enriched by one bubble a quadrilateral. The superconvergence in the pressure of the proposed pair is analyzed on uniform rectangular meshes, and tested numerically on uniform...

### A new $\theta$-scheme algorithm and incompressible FEM for viscoelastic fluid flows

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

### A novel approach to modelling of flow in fractured porous medium

Kybernetika

There are many problems of groundwater flow in a disrupted rock massifs that should be modelled using numerical models. It can be done via “standard approaches” such as increase of the permeability of the porous medium to account the fracture system (or double-porosity models), or discrete stochastic fracture network models. Both of these approaches appear to have their constraints and limitations, which make them unsuitable for the large- scale long-time hydrogeological calculations. In the article,...

### A numerical approximation of non-Fickian flows with mixing length growth in porous media.

Acta Mathematica Universitatis Comenianae. New Series

### A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems

Applications of Mathematics

A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation $-ϵ{u}^{n}+p{u}^{\text{'}}+qu=f$ are presented and analyzed theoretically. The positive number $ϵ$ is supposed to be much less than the discretization step and the values of $\left|p\right|,q$. An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.

### A piecewise P2-nonconforming quadrilateral finite element

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the...

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