Superconvergence And Error Estimates Of Discontinuous Galerkin Methods For Hyperbolic And Fourth-order Equations

Discontinuous Galerkin methods are a class of finite element methods originally de-vised to numerically solve hyperbolic conservation laws containing only first order spatialderivatives. As an extension of discontinuous Galerkin methods, the local discontinuousGalerkin methods are developed with the goal of solving high order partial diferential e-quations. Discontinuous Galerkin and local discontinuous Galerkin methods are recentlyproposed and developed high order accurate high resolution numerical methods not on-ly in obtaining arbitrary high order accuracy approximation to the exact solution withinsmooth regions but also in producing sharp and non-oscillatory discontinuity transitionsnear discontinuous solutions, including shocks and contact discontinuities, by employ-ing suitable limiters (for instance, weighted essentially non-oscillatory limiters that arerecently introduced and developed).Our attention is mainly paid to the superconvergence property and error estimatesof discontinuous Galerkin and local discontinuous Galerkin methods for solving severaltime-dependent partial diferential equations. An important motivation for investigatingsuch superconvergence is to lay a solid theoretical foundation for the fact that the errorbetween the discontinuous Galerkin solution and the exact solution does not grow over along time period. This property is especially prominent for fine meshes. The supercon-vergence property also demonstrates the capability of discontinuous Galerkin method inresolving waves. The superconvergence results presented in this work regarding the localdiscontinuous Galerkin method for a class of linear fourth-order time-dependent problemsand discontinuous Galerkin methods for nonlinear hyperbolic conservation laws broadenand reinforce the superconvergence theory of discontinuous Galerkin methods. Mean-while, error analysis of discontinuous Galerkin methods provides a solid theoretical foun-dation for its high order accuracy feature. Optimal a priori error estimates are obtainedfor discontinuous Galerkin methods applied to nonlinear hyperbolic conservation laws inmultiple dimensions, which further strengthens the convergence theory of discontinuousGalerkin methods.We start by exploring the superconvergence property of the local discontinuousGalerkin methods for solving one-dimensional linear fourth-order time-dependent prob-lems. By constructing a special global projection, the local discontinuous Galerkin so- lution is proved to be(k+3/2)th order superconvergent to a particular projection of theexact solution, when alternating numerical fluxes are used. Here and below, k≥1is thepiecewise polynomial degree of the finite element space. This result extends the work byCheng and Shu, in which superconvergence of the local discontinuous Galerkin methodapplied to one-dimensional linear convection-difusion equations is analyzed. Moreover,optimal convergence results on numerical solution, its spatial derivatives of diferent or-ders as well as its time derivative are obtained. Various numerical examples, includinglinear problems, initial boundary value problems, nonlinear equations and solutions hav-ing singularities, are reported to verify the superconvergence property and time evolutionof the local discontinuous Galerkin method.We proceed to propose and analyze the superconvergence property of discontinu-ous Galerkin methods for solving one-dimensional nonlinear time-dependent hyperbolicconservation laws. By virtue of the linearized approach based on Taylor expansions andan a priori assum(ption about the numerical solution, the discontinuous Galerkin solutionis proved to be(k+3/2)2th order superconvergent to a particular projection of the exact so-lution when upwind numerical fluxes are used. This work extends the superconvergenceanalysis of discontinuous Galerkin methods from linear problems to nonlinear ones. Inaddition, optimal error estimates on numerical solution and its time derivative are derived.A series of numerical experiments, including one-dimensional nonlinear conservationlaws with polynomial flux functions and strong nonlinearities as well as two-dimensionalcases, are presented to validate the superconvergence property and time evolution of thediscontinuous Galerkin method. The computational efciency of discontinuous Galerkinmethods for solving nonlinear conservation laws is thusly demonstrated.We end by studying the error estimates of discontinuous Galerkin methods for multi-dimensional nonlinear hyperbolic conservation laws on Cartesian meshes. By using su-perconvergence property of multi-dimensional special projections, optimal error estimatesof discontinuous Galerkin methods for tensor product polynomials of degree at most k(k≥2) are obtained when upwind numerical fluxes are considered. It is also pointed outthat extension of the current approach to proving suboptimal error estimates for generalmonotone numerical fluxes can be easily made.In summary, discontinuous Galerkin methods are a class of high order accurate highresolution numerical methods for solving hyperbolic conservation laws and convection-dominated partial diferential equations. The emphasis of this work is put on the study of superconvergence property and error estimates of discontinuous Galerkin and local dis-continuous Galerkin methods, which provides a solid theoretical foundation for the nicebehaviour of discontinuous Galerkin methods for long time simulations. The high orderaccuracy performance of the discontinuous Galerkin method is further confirmed. Erroranalysis together with numerical experiments shows the computational efciency and ca-pability of discontinuous Galerkin methods and local discontinuous Galerkin methods forsolving linear equations, nonlinear equations, one-and multi-dimensional time-dependentproblems.