Achieving stable, accurate, monotone and efficient (SAME) discrete approximate solution is of tremendous interest for convection-dominated computational fluid dynamics (CFD) applications. Recently, a non-linear SubGrid eMbedded (SGM) finite element basis was developed for generating multidimensional SAME solutions via the weak statement (WS). The theory confirms that only the Navier-Stokes diss...

Difficulties associated with free-surface finite element flow simulations are related to a) nonlinear and advective nature of most hydrodynamic flows, b) requirements for compatibility between velocity and pressure interpolation, c) maintaining a valid computational mesh in the presence of moving boundaries, and d) enforcement of the kinematic conditions at the free surface. Focusing on the las...

This artic!e considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. _Veighted residual approxim...

We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin method. The velocity (momentum) equation is approximated by a finite element method on div–curl form using the nonconforming Crouzeix– Raviart space. Our main result is that...

This paper deals with the convergence properties of the nonconforming quadrilateral Wilson element for a class of nonlinear parabolic problems in two space dimensions. Optimal H and L2 error estimates for the continuous time Galerkin approximations are derived. It is also shown for rectangular meshes that the gradient of the Wilson element solution possesses superconvergence, and that the Lx er...

We consider the viscous Burgers equation with a fractional nonlinear term as a model involving non-local nonlinearities in conservation laws, which, surprisingly, has an analytical solution obtained by a fractional extension of the Hopf-Cole transformation. We use this model and its inviscid limit to develop stable spectral and discontinuous Galerkin spectral element methods by employing the co...