We introduce a stochastic Galerkin mixed formulation of the steady-state diffusion equation and focus on the efficient iterative solution of the saddle-point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions. So-called mean-based preconditioners, based on fast solvers for scalar diffusion problems, are introduced for use ...

A linearized spectral Galerkin/finite difference approach is developed for variable fractional-order nonlinear diffusion–reaction equations with a fixed time delay. The temporal discretization the variable-order fractional derivative performed by L1-approximation. An appropriate basis function in terms of Legendre polynomials used to construct Galerkin method spatial second-order operator. main...

free vibration of sandwich plates with temperature dependent functionally graded (fg) face sheets in various thermal environments is investigated. the material properties of fg face sheets are assumed to be temperature-dependent and vary continuously through the thickness according to a power-law distribution in terms of the volume fractions of the constituents. also, the material properties of...

Error representation formulas and a posteriori error estimates for numerical solutions of hyperbolic conservation laws are considered with specialized variants given for the Godunov finite volume and discontinuous Galerkin finite element methods. The error representation formulas utilize the solution of a dual problem to capture the nonlocal error behavior present in hyperbolic problems. The er...

Time finite element methods are developed for the equations of structural dynamics. The approach employs the time-discontinuous Galerkin method and incorporates stabilizing terms having least-squares form. These enable a general convergence theorem to be proved in a norm stronger than the energy norm. Results are presented from finite difference analyses of the time-discontinuous Galerkin and l...

An adaptive discontinuous Galerkin finite element method for linear elasticity problems is presented. We develop an a posteriori error estimate and prove its robustness with respect to nearly incompressible materials (absence of volume locking). Furthermore, we present some numerical experiments which illustrate the performance of the scheme on adaptively refined meshes.

This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation. The weak Galerkin method, first introduced by two of the authors (J. Wang and X. Ye) in [52] for second order elliptic problems, is based on the concept of discrete weak gradients. The method uses completely discrete finite element functions and, using certain discrete spaces an...

We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially nonoscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by [18, 24], a general framework, for arbitrary order of accuracy, ...

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate meth...

We present an adaptive coupled level-set/volume-of-fluid (ACLSVOF) method for interfacial flow simulations on unstructured triangular grids. At each time step, we evolve both the level set function and the volume fraction. The level set function is evolved by solving the level set advection equation using a discontinuous Galerkin finite element method. The volume fraction advection is performed...