We summarize several techniques of analysis for finite element methods for linear hyperbolic problems, illustrating their key properties on the simplest model problem. These include the discontinuous Galerkin method, the continuous Galerkin methods on rectangles and triangles, and a nonconforming linear finite element on a special triangular mesh.

Abstract The parametric surrogate models for partial differential equations (PDEs) are a necessary component many applications in computational sciences, and the convolutional neural networks (CNNs) have proven to be an excellent tool generate these surrogates when fields present. CNNs commonly trained on labeled data based one-to-one sets of parameter-input PDE-output fields. Recently, residua...

This article deals with the computational study of the nonlinear Galerkin method, which is the extension of commonly known Faedo-Galerkin method. The weak formulation of the method is derived and applied to the particular ScottWang-Showalter reaction-diffusion model concerning the problem of combustion of hydrocarbon gases. The proof of convergence of the method based on the method of compactne...

In this paper, the well known iterated Galerkin method and iterated Galerkin-Kantorovich regularization method for approximating the solution of Fredholm integral equations of the second kind are generalized to Hammerstein equations with smooth and weakly singular kernels. The order of convergence of the Galerkin method and those of superconvergence of the iterated methods are analyzed. Numeric...

We consider a discontinuous Galerkin finite element method for the advection–reaction equation in two space–dimensions. For polynomial approximation spaces of degree greater than or equal to two on triangles we propose a method where stability is obtained by a penalization of only the upper portion of the polynomial spectrum of the jump of the solution over element edges. We prove stability in ...

A family of explicit space-time finite element methods for the initial boundary value problem for linear, symmetric hyperbolic systems of equations is described and analyzed. The method generalizes the discontinuous Galerkin method and, as is typical for this method, obtains error estimates of order O(hn+1/2) for approximations by polynomials of degree ≤ n.

In this paper we study numerically the KdV-top equation and compare it with the Boussinesq equations over uneven bottom. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa-Holm equat...

In this paper an implicit fully discrete local discontinuous Galerkin (LDG) finite element method is applied to solve the time-fractional seventh-order Korteweg-de Vries (sKdV) equation, which is introduced by replacing the integer-order time derivatives with fractional derivatives. We prove that our scheme is unconditional stable and L error estimate for the linear case with the convergence ra...

This work presents the coupling of two locally conservative methods for elliptic problems: namely, the discontinuous Galerkin method and the mixed finite element method. The couplings can be defined with or without interface Lagrange multipliers. The formulations are shown to be equivalent. Optimal error estimates are given; penalty terms may or may not be included. In addition, the analysis fo...