First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems

We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data f in L2 space. The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov - Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014), pp. 537-552]. This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods - numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform

Primal dual mixed finite element methods for indefinite advection--diffusion equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.Comment: 25 pages, 6 figures, 5 table

Stabilised finite element methods for ill-posed problems with conditional stability

In this paper we discuss the adjoint stabilised finite element method introduced in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on Scientific Computing, and how it may be used for the computation of solutions to problems for which the standard stability theory given by the Lax-Milgram Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to ill-posed problems that have some conditional stability property and prove (conditional) error estimates in an abstract framework. As a model problem we consider the elliptic Cauchy problem and provide a complete numerical analysis for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equation

A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes

In this paper, we propose a way to weakly prescribe Dirichlet boundary conditions in embedded finite element meshes. The key feature of the method is that the algorithmic parameter of the formulation which allows to ensure stability is independent of the numerical approximation, relatively small, and can be fixed a priori. Moreover, the formulation is symmetric for symmetric problems. An additional element‐discontinuous stress field is used to enforce the boundary conditions in the Poisson problem. Additional terms are required in order to guarantee stability in the convection–diffusion equation and the Stokes problem. The proposed method is then easily extended to the transient Navier–Stokes equations

A stabilised finite element method for the convection-diffusion-reaction equation in mixed form

This paper is devoted to the approximation of the convection-diffusion-reaction equation using a mixed, first-order, formulation. We propose, and analyse, a stabilised finite element method that allows equal order interpolations for the primal and dual variables. This formulation, reminiscent of the Galerkin least-squares method, is proven stable and convergent. In addition, a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out, and the different methods are compared in terms of accuracy, stability, and sharpness of the layers for two different classical test problems

A first order system least squares method for the Helmholtz equation

We present a first order system least squares (FOSLS) method for the Helmholtz equation at high wave number k, which always deduces Hermitian positive definite algebraic system. By utilizing a non-trivial solution decomposition to the dual FOSLS problem which is quite different from that of standard finite element method, we give error analysis to the hp-version of the FOSLS method where the dependence on the mesh size h, the approximation order p, and the wave number k is given explicitly. In particular, under some assumption of the boundary of the domain, the L2 norm error estimate of the scalar solution from the FOSLS method is shown to be quasi optimal under the condition that kh/p is sufficiently small and the polynomial degree p is at least O(\log k). Numerical experiments are given to verify the theoretical results

Mixed formulations for the convection-diffusion equation

This thesis explores the numerical stability of the stationary Convection-Diffusion-Reaction (CDR) equation in mixed form, where the second-order equation is expressed as two first-order equations using a second variable relating to a derivative of the primary variable. This first-order system uses either a total or diffusive flux formulation. Westart by numerically testing the unstabilised Douglas and Roberts classical discretisation of the mixed CDR equation using Raviart-Thomas elements. The results indicate that,as expected, for both total and diffusive flux, the stability of the formulation degrades dramatically as diffusion decreases.Next, we investigate stabilised formulations that are designed to improve the ability of the discrete problem to cope with problems containing layers. We test the Masud and Kwack method that uses Lagrangian elements but whose analysis has not been developed.We then significantly modify the formulation to allow us to prove existence of a solution and facilitate the analysis. Our new method, which uses total flux, is then tested for convergence with standard tests and found to converge satisfactorily over a range of values of diffusion.Another family of first-order methods called First-Order System of Least-Squares (FOSLS/LSFEM) is also investigated in relation to solving the CDR equation. These symmetric,elliptic methods do not require stabilisation but also do not cope well with sharp layers and small diffusion. Modifications have been proposed and this study includes aversion of Chen et al. which uses diffusive flux, imposing boundary conditions weakly in a weighted formulation.We test our new method against all the aforementioned methods, but we find that other methods do not cope well with layers in standard tests. Our method compares favourably with the standard Streamline-Upwind-Petrov-Galerkin method (SUPG/SDFEM), but overall is not a significant improvement. With further fine-tuning, our method could improve but it has more computational overhead than SUPG.This thesis explores the numerical stability of the stationary Convection-Diffusion-Reaction (CDR) equation in mixed form, where the second-order equation is expressed as two first-order equations using a second variable relating to a derivative of the primary variable. This first-order system uses either a total or diffusive flux formulation. Westart by numerically testing the unstabilised Douglas and Roberts classical discretisation of the mixed CDR equation using Raviart-Thomas elements. The results indicate that,as expected, for both total and diffusive flux, the stability of the formulation degrades dramatically as diffusion decreases.Next, we investigate stabilised formulations that are designed to improve the ability of the discrete problem to cope with problems containing layers. We test the Masud and Kwack method that uses Lagrangian elements but whose analysis has not been developed.We then significantly modify the formulation to allow us to prove existence of a solution and facilitate the analysis. Our new method, which uses total flux, is then tested for convergence with standard tests and found to converge satisfactorily over a range of values of diffusion.Another family of first-order methods called First-Order System of Least-Squares (FOSLS/LSFEM) is also investigated in relation to solving the CDR equation. These symmetric,elliptic methods do not require stabilisation but also do not cope well with sharp layers and small diffusion. Modifications have been proposed and this study includes aversion of Chen et al. which uses diffusive flux, imposing boundary conditions weakly in a weighted formulation.We test our new method against all the aforementioned methods, but we find that other methods do not cope well with layers in standard tests. Our method compares favourably with the standard Streamline-Upwind-Petrov-Galerkin method (SUPG/SDFEM), but overall is not a significant improvement. With further fine-tuning, our method could improve but it has more computational overhead than SUPG