Application of a Galerkin finite element method to atmospheric transport problems

Numerical simulation of the movement of a contaminant within the atmosphere presents difficulties due to (a) The multi-dimensionality of the problem; (b) The fact that the horizontal transport is usually convection dominated; (c) The boundary conditions are mixed; (d) Both slow and fast atmospheric chemical reactions can be important. In this study, numerical experiments using a Crank-Nicolson Galerkin finite element method to solve the time-dependent partial differential equations demonstrate the applicability and accuracy of this method for the variety of conditions encountered in atmospheric pollutant modeling. The Crank-Nicolson Galerkin method using piecewise linear, piecewise cubic Hermite polynomials, and upwind finite elements is shown to accurately model the pure convection of initial wave forms. Numerical results studying the interactions of convection, diffusion, chemical reaction, pollutant removal, and the effects of contaminant emission source strength, source location and multiple sources are also presented.     

Galerkin methods and L2-error estimates for hyperbolic integro-differential equations

In this paper we shall study Galerkin approximations to the solution of linear second-order hyperbolic integro-differential equations. The continuous and Crank-Nicolson discrete time Galerkin procedures will be defined and optimal error estimates for these procedures are demonstrated by using a “non-classical” elliptic projection.     

Investigation of some finite-difference techniques for solving the boundary layer equations

The Levy-Lees form of the laminar boundary layer equations is solved with several second-order accurate finite-difference schemes. For incompressible flow the methods investigated include three forms of the Crank-Nicolson scheme, four variations of the Keller box scheme and a modified box scheme. The number of iterations required at each step along the surface to obtain a second-order accurate scheme is studied. The accuracy of the schemes with various step-sizes is determined for the boundary layer flow with a linearly retarded edge velocity. In addition, one form of the Crank-Nicolson scheme is extended to compressible flows and its accuracy and behavior are also examined for the linearly retarded flow case. The results of this investigation show that the coupled continuity-momentum form of the Crank-Nicolson scheme is second-order with one iteration at each step and requires less time than the Keller box scheme.     

Non-classicalH 1 projection and Galerkin methods for non-linear parabolic integro-differential equations

In this paper the Galerkin method is analyzed for the following nonlinear integro-differential equation of parabolic type: $$c(u)u_t = \nabla \cdot \{ a(u)\nabla u + \int_0^t {b(x, t, r, u(x, r))} \nabla u(x, r) dr\} + f (u)$$ Optimal L ² error estimates for Crank-Nicolson and extrapolated Crank-Nicolson approximations are derived by using a non-classicalH ¹ projection associated with the above equation. Both schemes result in procedures which are second order correct in time, but the latter requires the solution of a linear algebraic system only once per time step.     

The characteristic-Galerkin method for advection-dominated problems—An assessment

In this note we present results of an accuracy analysis of a recent characteristic-based Galerkin method suited for advection-dominated problems. The analysis shows that the numerical propagation characteristics of the explicit time-stepping scheme which uses linear basis functions for spatial discretization are superior to those of the related classical Lax-Wendroff method and the implicit Crank-Nicolson scheme. The model is subjected to three analytical test problems which embrace many essential realistic features of environmental and coastal hydrodynamic applications: pure advection of a steep Gaussian profile, dispersion of a continuous source in an oscillating flow, and long-wave propagation with bottom frictional dissipation in a rectangular channel. The numerical results demonstrate that the accuracy achieved with the present scheme is excellent and comparable to that of a characteristic-based finite difference scheme which uses Hermitian cubic interpolating polynomials. The results reported herein suggest strongly further use and testing of this robust model in engineering practice.     

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.     

Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

This work focuses on simultaneous approximation terms (SATs) for multidimensional summation-by-parts (SBP) discretizations of linear second-order partial differential equations with variable coefficients. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties for general operators, including those operators that do not have nodes on their boundary or do not correspond with a collocation discontinuous Galerkin method. Based on these conditions, we generalize the modified scheme of Bassi and Rebay and the symmetric interior penalty Galerkin method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.     

A collocation solution for Burgers' equation using cubic B-spline finite elements

The non-linear Burgers' equation is solved numerically by a B-spline finite element method. The approach used is based on collocation of cubic B-splines over finite elements so that we have continuity of the dependent variable and its first two derivatives throughout the solution range. A linear stability analysis shows that a numerical scheme based on a Crank-Nicolson approximation in time is unconditionally stable. Three standard problems are used to validate the algorithm. Comparisons are made with published numeric and analytic solutions. The proposed method performs well.     

Boundary element method for membrane and torsion crack problems

Here we present as an application of [12] an improved Galerkin method for the boundary integral equations governing a plane interface problem. Membrane and torsion crack problems can be treated by slight modifications.A tedious analysis incorporating the Mellin transform shows that the coupled system of integral equations—with some Fredholm equations of the second kind and some of the first kind—on the boundary curve Γ is strongly elliptic, i.e., there holds a Gårding inequality. This property implies convergence of almost optimal order of the Galerkin procedure. The use of singularity functions together with regular finite elements on Γ provides convergence results in a scale of Sobolev spaces and even quasi-optimal asymptotic error estimates for the stress intensity factors. These factors are computed directly by our Galerkin scheme, i.e., no additional computations are needed.The Galerkin method is implemented by an appropriate numerical integration leading to a Galerkin collocation. The latter is a modified collocation method which can be easily implemented on computing machines.     

Adaptive approximations for 3-D electrostatic plate problems

The solution of a weakly singular integral equation on a plane surface piece Γ is approximated via the Galerkin method, using piecewise constant elements. The determination of the solution of this integral equation (with the single layer potential) is a classical problem in physics, since its solution represents the charge density of a thin, electrified plate Γ loaded with a given potential. The capacitance of Γ is proportional to the integral of the charge density. Within, two adaptive strategies are presented, which based upon an approximation of the local residue, refine the grid locally. Numerical results are given for the unknown capacitance which indicate an exponential rate of convergence of the boundary element Galerkin method.     

A finite element first-order equation formulation for the small-disturbance transonic flow problem

The nonlinear, mixed elliptic hyperbolic equation describing a steady transonic flow is considered. The original equation is replaced by a system of first-order equations that are hyperbolic in time and defined in terms of velocity components. Parabolic regularization terms are added to capture shock wave solutions and to damp iterative solution algorithms. A finite element Galerkin method in space and a Crank-Nicolson finite difference method in iterative time are used to reduce the problem to the solution of a system of algebraic equations. Stability and convergence characteristics of the iterative method are discussed. The numerical implementation of the method is explained, and numerical results are presented.     

Weak Galerkin method for the Biot’s consolidation model

We develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure without special treatment. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.     

A review of some numerical methods for reaction-diffusion equations

Some fixed-node finite-difference schemes and a finite element method are applied to a reaction-diffusion equation which has an exact traveling wave solution. The accuracy of the methods is assessed in terms of the computed steady state wave speed which is compared with the exact speed. The finite element method uses a semi-discrete Galerkin approximation. The finite-difference schemes discussed in this review include two explicit algorithms, three methods of lines, two implicit procedures, two majorant operator-splitting techniques, four time-linearization schemes and the Crank-Nicolson method. The effects of the truncation errors and linearization on the computed wave speed are determined. The application of these techniques to reaction-diffusion equations appearing in combustion theory is also discussed. The review is limited to fixed-node techniques and does not include moving or adaptive finite-difference and adaptive finite element methods.     

Error Analysis of Chebyshev-Legendre Pseudo-spectral Method for a Class of Nonclassical Parabolic Equation

Many physical phenomena are modeled by nonclassical parabolic initial boundary value problems which involve a nonclassical term u _xxt in the governed equation. Combining with the Crank-Nicolson/leapfrog scheme in time discretization, Chebyshev-Legendre pseudo-spectral method is applied to space discretization for numerically solving the nonclassical parabolic equation. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. The convergence rate shows ??spectral accuracy??. Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.     

A posteriori error estimates for fem–bem couplings of three-dimensional electromagnetic problems

We present residual based and p-hierarchical a posteriori error estimators for a Galerkin method coupling finite elements and boundary elements for time–harmonic interface problems in electromagnetics; special emphasis is taken for the eddy current problem. The Galerkin discretization uses lowest order Nédélec elements in the interior domain and vectorial surface rotations of continuous, piecewise linear functions on the interface boundary. Singular, weakly singular and hypersingular boundary integral operators appearing in the variational formulation show up in the terms of the error estimators as well. The estimators are derived from the defect equation using Helmholtz and Hodge decompositions. Numerical tests underline reliability and efficiency of the given error estimators yielding reasonable mesh refinements.     

A Fast Solver for Boundary Integral Equations of the Modified Helmholtz Equation

The main purpose of this paper is to develop a fast fully discrete Fourier–Galerkin method for solving the boundary integral equations reformulated from the modified Helmholtz equation with boundary conditions. We consider both the nonlinear and the Robin boundary conditions. To tackle the difficulties caused by the two types of boundary conditions, we provide an improved version of the Galerkin method based on the Fourier basis. By employing a matrix compression strategy and efficient numerical quadrature schemes for oscillatory integrals, we obtain fully discrete nonlinear or linear system. Finally, we use the multilevel augmentation method to solve the resulting systems. We point out that the proposed method enjoys an optimal convergence order and a nearly linear computational complexity. The theoretical estimates are confirmed by the performance of this method on several numerical examples.     

A lumped Galerkin method for solving the Burgers equation

A numerical solution of the one-dimensional Burgers equation is obtained using a lumped Galerkin method with quadratic B-spline finite elements. The scheme is implemented to solve a set of test problems with known exact solutions. Results are compared with published numerical and exact solutions. The proposed scheme performs well. A linear stability analysis shows the scheme to be unconditionally stable.     

Enriched finite element subspaces for dual–dual mixed formulations in fluid mechanics and elasticity

In this paper we unify the derivation of finite element subspaces guaranteeing unique solvability and stability of the Galerkin schemes for a new class of dual-mixed variational formulations. The approach, which has been applied to several linear and nonlinear boundary value problems, is based on the introduction of additional unknowns given by the flux and the gradient of velocity, and by the stress and strain tensors and rotations, for fluid mechanics and elasticity problems, respectively. In this way, the procedure yields twofold saddle point operator equations as the resulting weak formulations (also named dual–dual ones), which are analyzed by means of a slight generalization of the well known Babuška–Brezzi theory. Then, in order to introduce well posed Galerkin schemes, we extend the arguments used in the continuous case to the discrete one, and show that some usual finite elements need to be suitably enriched, depending on the nature of the problem. This leads to piecewise constant functions, Raviart–Thomas of lowest order, PEERS elements, and the deviators of them, as the appropriate subspaces.     

A Central Discontinuous Galerkin Method for Hamilton-Jacobi Equations

In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of Hamilton-Jacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since Hamilton-Jacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a “weighted-residual” or “stabilization-based” formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for Hamilton-Jacobi equations. The L ² stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general Hamilton-Jacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.