Streamline Diffusion Method Research Articles

Applications of the Newton-Raphson method in a SDFEM for inviscid Burgers equation

In this paper we investigate in detail the applications of the classical Newton-Raphson method in connection with a space-time finite element discretization scheme for the inviscid Burgers equation in one dimensional space. The underlying discretization method is the so-called streamline diffusion method, which combines good stability properties with high accuracy. The coupled nonlinear algebraic equations thus obtained in each space-time slab are solved by the generalized Newton-Raphson method. Exploiting the band-structured properties of the Jacobian matrix, two different algorithms based on the Newton-Raphson linearization are proposed. In a series of examples, we show that in each time-step a quadratic convergence order is attained when the Newton-Raphson procedure applied to the corresponding system of nonlinear equations.

A cascadic multigrid asymptotic-preserving discrete ordinate discontinuous streamline diffusion method for radiative transfer equations with diffusive scalings

In this paper, we develop a cascadic multigrid asymptotic-preserving discrete ordinate discontinuous streamline diffusion scheme for radiative transfer equations (RTE) with multiple scalings. Our new method employs a simple and efficient cascadic multigrid method to the discretized RTE system as well as a diffusion synthetic acceleration technique as an efficient smoother to accelerate the convergence of the iteration in diffusive region. Furthermore, by applying the discontinuous streamline diffusion schemes, improved convergence condition in heterogeneous media and the asymptotic-preserving (AP) property can be achieved. The AP property of these methods will be explained formally and demonstrated numerically. Numerical results are presented to show the effectiveness and efficiency of the proposed numerical scheme for solving radiative transfer equations, especially in diffusive and heterogeneous media.

A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces

We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included.

A gradient recovery–based adaptive finite element method for convection‐diffusion‐reaction equations on surfaces

SummaryIn this paper, we present an adaptive mesh refinement method for solving convection‐diffusion‐reaction equations on surfaces, which is a fundamental subproblem in many models for simulating the transport of substances on biological films and solid surfaces. The method considered is a combination of well‐known techniques: the surface finite element method, streamline diffusion stabilization, and the gradient recovery–based Zienkiewicz‐Zhu error estimator. The streamline diffusion method overcomes the instability issue of the finite element method for the dominance of the convection. The gradient recovery–based adaptive mesh refinement strategy enables the method to provide high‐resolution numerical solutions by relatively fewer degrees of freedom. Moreover, the implementation detail of a surface mesh refinement technique is presented. Various numerical examples, including the convection‐dominated diffusion problems with large variations of solutions, nearly singular solutions, discontinuous sources, and internal layers on surfaces, are presented to demonstrate the efficacy and accuracy of the proposed method.

Adaptive Concepts for Stochastic Partial Differential Equations

An adaptive time stepping method to numerically provide approximate weak approximations of solutions of general SPDEs is proposed, where local step sizes are chosen in regard of the distance between empirical laws of subsequent time iterates and extrapolated data. The histogram-based estimator uses a data-driven partitioning of the high-dimensional state space, and efficient sampling by bootstrapping. Time adaptivity is then complemented by a local refinement/coarsening strategy of the spatial mesh of a stochastic version of the ZZ-estimator. Next to an improved accuracy, we observe a significantly reduced empirical variance of standard estimators, and therefore a reduced sampling effort. The performance of the adaptive strategies is studied for SPDEs with linear drift, including the convection-dominated case where the streamline diffusion method is adopted to attain a stable discretization, and the stochastic version of the non-linear harmonic map heat flow to the sphere $${\mathbb {S}}^{2}$$ where approximate solutions exhibit discrete blow-up dynamics.

Two types of spurious oscillations at layers diminishing methods for convection–diffusion–reaction equations on surface

In this article, we consider the steady convection–diffusion–reaction equation posed on general surface in convection-dominated regimes, which is an important problem in many models for simulating substances transport on ultrathin material and solid surfaces. Two types of spurious oscillations at layers diminishing (SOLD) methods are considered when the streamline diffusion method cannot completely eliminates the spurious oscillations at layers for convection-dominated problem with nonsmooth solution. One is the edge stabilization method that uses least square stabilization of the gradient jumps a across element boundaries, the other, a Petrov–Galerkin method, is called the Mizukami–Hughes method. An error analysis of the edge stabilization method on surface is provided. Finally, numerical experiments including the comparisons among these methods are presented to illustrate the accuracy and efficiency of the SOLD methods.

A new streamline diffusion finite element method for the generalized Oseen problem

This paper aims to present a new streamline diffusion method with low order rectangular Bernardi-Raugel elements to solve the generalized Oseen equations. With the help of the Bramble-Hilbert lemma, the optimal errors of the velocity and pressure are estimated, which are independent of the considered parameter e. With an interpolation postprocessing approach, the superconvergent error of the pressure is obtained. Finally, a numerical experiment is carried out to confirm the theoretical results.

On Multiscale Variational and Streamline Diffusion Schemes for a Coupled Nonlinear Telegraph System

ABSTRACTWe propose a hybrid of streamline diffusion (SD) method and variational multiscale scheme (VMS) for approximate solution of a coupled nonlinear system of telegraph equations. The reason for using multiscale scheme is due to the fact that, compared to the evolved system, in the primary time iteration steps higher degree schemes in spatial variable are not necessary. On the other hand, the multiscale strategy may be a viewed as a particular type of adaptivity where different scales play the role of coarse or fine refinement procedures. In this setting, certain data and geometric singularities that are better studied through an adaptive approach, which here is replaced by a multiscale scheme. We prove stability estimates and derive optimal convergence rates due to the maximal available regularity of the exact solution. This study concerns both theoretical as well as some numerical aspects. The theoretical part, mainly, concerns the stability and convergence issues whereas in the numerical part, we deal with the construction and of the discretization multiscale schemes. The results are justified through some numerical implementations where, in particular by the constructed multiscale scheme, one may circumvent the above mentioned trouble with the primary time steps.

Adaptive Finite Element Method for Steady Convection-Diffusion Equation

This paper examines the numerical solution of the convection-diffusion equation in 2-D. The solution of this equation possesses singularities in the form of boundary or interior layers due to non-smooth boundary conditions. To overcome such singularities arising from these critical regions, the adaptive finite element method is employed. This scheme is based on the streamline diffusion method combined with Neumann-type posteriori estimator. The effectiveness of this approach is illustrated by different examples with several numerical experiments.

Streamline-diffusion method of a lowest order nonconforming rectangular finite element for convection-diffusion problem

The streamline-diffusion method of the lowest order nonconforming rectangular finite element is proposed for convection-diffusion problem. By making full use of the element’s special property, the same convergence order as the previous literature is obtained. In which, the jump terms on the boundary are added to bilinear form with simple user-chosen parameter δK which has nothing to do with perturbation parameter e appeared in the problem under considered, the subdivision mesh size hK and the inverse estimate coefficient μ in finite element space.

Robusta posteriorierror estimates for stabilized finite element methods

There is a wide range of stabilized finite element methods for stationary and non-stationary convection-diffusion equations such as streamline diffusion methods, local projection schemes, subgrid-scale techniques, and continuous interior penalty methods to name only a few. We show that all these schemes give rise to the same robust a posteriori error estimates, i.e. the multiplicative constants in the upper and lower bounds for the error are independent of the size of the convection or reaction relative to the diffusion. Thus, the same error indicator can be used modulo higher order terms caused by data approximation.

Local projection stabilized method on unsteady Navier–Stokes equations with high Reynolds number using equal order interpolation

In this paper we propose and analyze a stabilized method for unsteady Navier–Stokes equations with high Reynolds number, using local projection stabilized method to control spurious oscillations in the velocities due to dominant convection, or in the pressure due to the velocity–pressure coupling. Using equal-order conforming elements in space and Crank–Nicolson difference in time, we derive a fully discrete formulation. We prove stability and convergence of the approximate solution. The error estimates hold irrespective of the Reynolds number, provided the exact solution is smooth. This result is comparable with the streamline diffusion and continuous interior penalty methods.

Discrete-Ordinates and Streamline Diffusion Methods for a Flow Described by BGK Model

A rarefied gas flow through a channel with arbitrary cross section is studied based on the linearized Bhatnagar-Gross-Krook model. The discrete velocity and streamline diffusion finite element methods are combined to yield a numerical scheme. For this scheme we derive stability and optimal convergence rates in the L-2-type norms. The optimality is due to the maximal available regularity of the exact solution for the corresponding hyperbolic PDE. The potential of the proposed, combined methods is illustrated with some numerical examples.

Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations

This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart (CR) element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient.

Finite Element Computation of KPP Front Speeds in Cellular and Cat’s Eye Flows

We compute the front speeds of the Kolmogorov-Petrovsky-Piskunov (KPP) reactive fronts in two prototypes of periodic incompressible flows (the cellular flows and the cat's eye flows). The computation is based on adaptive streamline diffusion methods for the advection-diffusion type principal eigenvalue problem associated with the KPP front speeds. In the large amplitude regime, internal layers form in eigenfunctions. Besides recovering known speed growth law for the cellular flow, we found larger growth rates of front speeds in cat's eye flows due to the presence of open channels, and the front speed slowdown due to either zero Neumann boundary condition at domain boundaries or frequency increase of cat's eye flows.

The characteristic finite difference streamline diffusion method for convection-dominated diffusion problems

In this paper, we consider the characteristic finite difference streamline diffusion method for two-dimensional convection-dominated diffusion problems. The scheme is combined the method of characteristics with the finite difference streamline diffusion (FDSD) method to create the characteristic FDSD (C-FDSD) procedures. Stability analysis and error estimate of the C-FDSD method are deduced. The scheme not only realizes the purpose of lowering the time-truncation error, using larger time step for solving the convection-dominated diffusion problems, but also keeps the favorable stability and high precision of the FDSD method. Finally, numerical experiments are presented to illustrate the availability of the scheme.

Parallel Galerkin domain decomposition procedures based on the streamline diffusion method for convection–diffusion problems

Based upon the streamline diffusion method, parallel Galerkin domain decomposition procedures for convection–diffusion problems are given. These procedures use implicit method in the sub-domains and simple explicit flux calculations on the inter-boundaries of sub-domains by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the parallelism can be achieved by these procedures. The explicit nature of the flux calculations induces a time step limitation that is necessary to preserve stability. Artificial diffusion parameters δ are given. By analysis, optimal order error estimate is derived in a norm which is stronger than L 2 -norm for these procedures. This error estimate not only includes the optimal H 1 -norm error estimate, but also includes the error estimate along the streamline direction ‖ β ⋅ ∇ ( u − U ) ‖ , which cannot be achieved by standard finite element method. Experimental results are presented to confirm theoretical results.

A Posteriori Error Estimate for Streamline Diffusion Method in Soving a Hyperbolic Equation

In this article, we use streamline diffusion method for the linear second order hyperbolic initial-boundary value problem. More specifically, we prove a posteriori error estimates for this method for the linear wave equation. We observe that this error estimates make finite element method increasingly powerful rather than other methods.

A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation

Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size h and polynomial degree p, and outperforms the standard upwind DG method.

A discontinuous finite difference streamline diffusion method for time-dependent hyperbolic problems

In this article, a new finite element method, discontinuous finite difference streamline diffusion method (DFDSD), is constructed and studied for first-order linear hyperbolic problems. This method combines the benefit of the discontinuous Galerkin method and the streamline diffusion finite element method. Two fully discrete DFDSD schemes (Euler DFDSD and Crank–Nicolson (CN) DFDSD) are constructed by making use of the difference discrete method for time variables and the discontinuous streamline diffusion method for space variables. The stability and optimal L 2 norm error estimates are established for the constructed schemes. This method makes contributions to the discontinuous methods. Finally, a numerical example is provided to show the benefit of high efficiency and simple implementation of the schemes.