Numerical implementation of a mixed finite element formulation for convection-diffusion problems

This document aims to the numerical solution of convection-diffusion problems in a fluid dynamics context by means of the Finite Element Method (FEM). It describes the classical finite element solution of convection-diffusion problems and presents the implementation and validation of a new formulation for improving the accuracy of the standard approach. On first place, the importance and need of numerical convection-diffusion models for Computational Fluid Dynamics (CFD) is emphasized, highlighting the similarities between the convection-diffusion equation and the governing equations of fluid dynamics for incompressible flow. The basic aspects of the finite element method needed for the standard solution of general convection-diffusion problems are then explained and applied to the steady state case. These include the weak formulation of the initial boundary value problem for the convection-diffusion equation and the posterior finite element spatial discretization of the weak form based on the Galerkin method. After their application to the steady transport equation a simple numerical test is performed to show that the standard Galerkin formulation is not stable in convection-dominated situations, and the need for stabilization is justified. Attention is then focused on the analysis of the truncation error provided by the Galerkin formulation, leading to the derivation of a classical stabilization technique based on the addition of artificial diffusion along the flow direction, the so-called streamline-upwind (SU) schemes. Next, a more general and modern stabilization approach known as the Sub-Grid-Scale (SGS) method is described, showing that SU schemes are a particular case of it. Taking into account all the concepts explained, a new mixed finite element formulation for convection-diffusion problems is presented. It has been proposed by Dr. Riccardo Rossi, a researcher from the International Center for Numerical Methods in Engineering (CIMNE), and consists on extending the original convection-diffusion equation to a system in mixed form in which both the unknown variable and its gradient are computed simultaneously, leading to an increase in the convergence rate of the solution. The formulation, which had not been tested before, is then implemented and validated by means of a multiphysics finite element software called \texttt{Kratos}. Eventually, the obtained results are analyzed, showing the improved performance of the mixed formulation in pure diffusion problems

Matrix-equation-based strategies for convection-diffusion equations

We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on the matrix equation formulation of the problem are proposed, which naturally approximate the original discretized problem. For certain types of convection coefficients, we show that the explicit solution of the matrix equation can effectively replace the linear system solution. Numerical experiments with data stemming from two and three dimensional problems are reported, illustrating the potential of the proposed methodology

Fast iterative solvers for convection-diffusion control problems

In this manuscript, we describe effective solvers for the optimal control of stabilized convection-diffusion problems. We employ the local projection stabilization, which we show to give the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the �first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to demonstrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the mesh size h, and the regularization parameter β, for a range of problems

New numerical approaches for modeling thermochemical convection in a compositionally stratified fluid

Seismic imaging of the mantle has revealed large and small scale heterogeneities in the lower mantle; specifically structures known as large low shear velocity provinces (LLSVP) below Africa and the South Pacific. Most interpretations propose that the heterogeneities are compositional in nature, differing in composition from the overlying mantle, an interpretation that would be consistent with chemical geodynamic models. Numerical modeling of persistent compositional interfaces presents challenges, even to state-of-the-art numerical methodology. For example, some numerical algorithms for advecting the compositional interface cannot maintain a sharp compositional boundary as the fluid migrates and distorts with time dependent fingering due to the numerical diffusion that has been added in order to maintain the upper and lower bounds on the composition variable and the stability of the advection method. In this work we present two new algorithms for maintaining a sharper computational boundary than the advection methods that are currently openly available to the computational mantle convection community; namely, a Discontinuous Galerkin method with a Bound Preserving limiter and a Volume-of-Fluid interface tracking algorithm. We compare these two new methods with two approaches commonly used for modeling the advection of two distinct, thermally driven, compositional fields in mantle convection problems; namely, an approach based on a high-order accurate finite element method advection algorithm that employs an artificial viscosity technique to maintain the upper and lower bounds on the composition variable as well as the stability of the advection algorithm and the advection of particles that carry a scalar quantity representing the location of each compositional field. All four of these algorithms are implemented in the open source FEM code ASPECT

Numerical Methods for Solving Convection-Diffusion Problems

Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. To handle flows in porous media, the fundamental issue is to model correctly the convective transport of individual phases. Moreover, for compressible media, the pressure equation itself is just a time-dependent convection-diffusion equation. For different problems, a convection-diffusion equation may be be written in various forms. The most popular formulation of convective transport employs the divergent (conservative) form. In some cases, the nondivergent (characteristic) form seems to be preferable. The so-called skew-symmetric form of convective transport operators that is the half-sum of the operators in the divergent and nondivergent forms is of great interest in some applications. Here we discuss the basic classes of discretization in space: finite difference schemes on rectangular grids, approximations on general polyhedra (the finite volume method), and finite element procedures. The key properties of discrete operators are studied for convective and diffusive transport. We emphasize the problems of constructing approximations for convection and diffusion operators that satisfy the maximum principle at the discrete level --- they are called monotone approximations. Two- and three-level schemes are investigated for transient problems. Unconditionally stable explicit-implicit schemes are developed for convection-diffusion problems. Stability conditions are obtained both in finite-dimensional Hilbert spaces and in Banach spaces depending on the form in which the convection-diffusion equation is written

Numerical solving unsteady space-fractional problems with the square root of an elliptic operator

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.Comment: 21 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1412.570

Low-diffusivity scalar transport using a WENO scheme and dual meshing

Interfacial mass transfer of low-diffusive substances in an unsteady flow environment is marked by a very thin boundary layer at the interface and other regions with steep concentration gradients. A numerical scheme capable of resolving accurately most details of this process is presented. In this scheme, the fourth-order accurate WENO method developed by Liu et al. (1994) was implemented on a non-uniform staggered mesh to discretize the scalar convection while for the scalar diffusion a fourth-order accurate central discretization was employed. The discretization of the scalar convection-diffusion equation was combined with a fourth-order Navier-Stokes solver which solves the incompressible flow. A dual meshing strategy was employed, in which the scalar was solved on a finer mesh than the incompressible flow. The solver was tested by performing a number of two-dimensional simulations of an unstably stratified flow with low diffusivity scalar transport. The unstable stratification led to buoyant convection which was modelled using a Boussinesq approximation with a linear relationship between flow temperature and density. The order of accuracy for one-dimensional scalar transport on a stretched and uniform grid was also tested. The results show that for the method presented above a relatively coarse mesh is sufficient to accurately describe the fluid flow, while the use of a refined mesh for the low-diffusive scalars is found to be beneficial in order to obtain a highly accurate resolution with negligible numerical diffusion