We study in this paper new developments of the Lagrange-Galerkin method for the advection equation. In the first part of the article we present a new improved error estimate of the conventional Lagrange-Galerkin method. In the second part, we introduce a new local projection stabilized Lagrange-Galerkin method, whereas in the third part we introduce and analyze a discontinuity-capturing Lagrange-Galerkin method. Also, attention has been paid to the influence of the quadrature rules on the stability and accuracy of the methods via numerical experiments.
We devise and analyze a class of the primal discontinuous Galerkin methods for the magnetic advection-diffusion problems based on the weighted-residual approach. In addition to the upwind stabilization, we find a new mechanism under the vector case that provides more flexibility in constructing the schemes. For the more general Friedrichs system, we show the stability and optimal error estimate, which boil down to two core ingredients -- the weight function and the special projection -- that contain information of advection. Numerical experiments are provided to verify the theoretical results.
In this paper we investigate the numerical approximation of the fractional diffusion, advection, reaction equation on a bounded interval. Recently the explicit form of the solution to this equation was obtained. Using the explicit form of the boundary behavior of the solution and Jacobi polynomials, a Petrov-Galerkin approximation scheme is proposed and analyzed. Numerical experiments are presented which support the theoretical results, and demonstrate the accuracy and optimal convergence of the approximation method.
In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable coefficient, which makes both the mathematical and numerical analysis challenging. To resolve this issue, we design appropriate test and trial functions to prove the inf-sup condition of the variable coefficient fractional diffusion, advection, reaction operators in suitable function spaces. Based on this property, we prove the well-posedness and regularity of the sol…
Time-space adaptive discontinuous Galerkin method for advection-diffusion equations with non-linear reaction mechanism
Middle East Technical University（中东理工大学）；Institute of Applied Mathematics, Middle East Technical University（中东理工大学应用数学研究所）；Department of Mathematics, Middle East Technical University（中东理工大学数学系）
We propose and analyze a pressure-stabilized projection Lagrange--Galerkin scheme for the transient Oseen problem. The proposed scheme inherits the following advantages from the projection Lagrange--Galerkin scheme. The first advantage is computational efficiency. The scheme decouples the computation of each component of the velocity and pressure. The other advantage is essential unconditional stability. Here we also use the equal-order approximation for the velocity and pressure, and add a symmetric pressure stabilization term. This enriched pressure space enables us to obtain accurate soluti…
Upwinded mass fluxes are described and analysed for advection operators discretised using mixed mimetic spectral elements. This involves a Petrov-Galerkin formulation by which the mass flux test functions are evaluated at downstream locations along velocity characteristics. As for the original mixed mimetic spectral element advection operator, the upwinded mass flux advection operator is conservative, however unlike the original advection operator, which is purely hyperbolic, the upwinded advection operator adds dissipation which is biased towards high wave numbers. The upwinded advection oper…
In this work, we introduce a generalization of the enriched Galerkin (EG) method. The key feature of our scheme is an adaptive two-mesh approach that, in addition to the standard enrichment of a conforming finite element discretization via discontinuous degrees of freedom, allows to subdivide selected (e.g. troubled) mesh cells in a non-conforming fashion and to use further discontinuous enrichment on this finer submesh. We prove stability and sharp a priori error estimates for a linear advection equation by using a specially tailored projection and conducting some parts of a standard converge…
In this paper, we use Fourier analysis to study the superconvergence of the semi-discrete discontinuous Galerkin method for scalar linear advection equations in one spatial dimension. The error bounds and asymptotic errors are derived for initial discretization by $L_2$ projection, Gauss-Radau projection, and other projections proposed by Cao et. al. For pedagogical purpose, the errors are computed in two different ways. In the first approach, we compute the difference between the numerical solution and a special interpolation of the exact solution, and show that it consists of an asymptotic e…
This is the second in a series of papers on implementing a discontinuous Galerkin (DG) method as an open source Matlab / GNU Octave toolbox. The intention of this ongoing project is to offer a rapid prototyping package for application development using DG methods. The implementation relies on fully vectorized matrix / vector operations and is comprehensively documented. Particular attention was paid to maintaining a direct mapping between discretization terms and code routines as well as to supporting the full code functionality in GNU Octave. The present work focuses on a two-dimensional time…
In this work, we consider a first order mean field games system with non-local couplings. A Lagrange-Galerkin scheme for the continuity equation, coupled with a semi-Lagrangian scheme for the Hamilton-Jacobi-Bellman equation, is proposed to discretize the mean field games system. The convergence of solutions to the scheme towards a solution to the mean field game system is established in arbitrary space dimensions. The scheme is implemented to approximate two mean field games systems in dimension one and two.
Insights on aliasing driven instabilities for advection equations with application to Gauss-Lobatto discontinuous Galerkin methods
Department of Mathematics, The Florida State University（佛罗里达州立大学数学系）；ETSIAE-UPM - Universidad Politécnica de Madrid. School of Aeronautics（马德里理工大学航空工程学院）
In this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell). We demonstrate, by carrying out numerical experiments, that the worst case error estimates given in the literature provide a good explanation for the error propagati…
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme
Institute of Mathematics, University of Mainz（美因茨大学数学研究所）；Faculty of Mathematics and Physics, Kanazawa University（金泽大学理学研究科）；Japan Science and Technology Agency (JST), PRESTO（日本科学技术振兴机构）；Department of Mathematics, Waseda University（早稻田大学数学系）
We study the convergence of the discontinuous Galerkin (DG) method applied to the advection-reaction equation on meshes with reentrant faces. On such meshes, the upwind numerical flux is not smooth, and so the numerical integration of the resulting face terms can only be expected to be first-order accurate. Despite this inexact integration, we prove that the DG method converges with order $\mathcal{O}(h^{p+1/2})$, which is the same rate as in the case of exact integration. Consequently, specialized quadrature rules that accurately integrate the non-smooth numerical fluxes are not required for …
A modified equation analysis for immersed boundary methods based on volume penalization: applications to linear advection-diffusion and high-order discontinuous Galerkin schemes
This study presents a two-step Lagrange-Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC). Firstly, the experimental order of convergence of the scheme is shown to see the second-order accuracy in time. Secondly, the effect of the TBC on a simple domain is discussed; the artificial reflections are kept from the Dirichlet boundaries and removed significantly from the transmission boundaries. Thirdly, the scheme is applied to a complex practical domain, i.e., the Bay of Bengal region, which is non-convex and includes islands. The effect of the TBC is di…
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part I: A nonlinear scheme
Institute of Mathematics, University of Mainz（美因茨大学数学研究所）；Faculty of Mathematics and Physics, Kanazawa University（金泽大学理学院）；Japan Science and Technology Agency (JST), PRESTO（日本科学技术振兴机构 PRESTO）；Department of Mathematics, Waseda University（早稻田大学数学系）
A mass-preserving two-step Lagrange-Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of $L^2$-theory. The introduced scheme maintains the advantages of the Lagrange-Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of seco…
We analyze families of primal high-order hybridizable discontinuous Galerkin (HDG) methods for solving degenerate (second-order) elliptic problems. One major trouble regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptic in the rest. This problem is thus quite delicate to solve since the exact solution is discontinuous at interfaces separating both elliptic and hyperbolic parts. The proposed HDG method is developed in a unified and compact …
