We present a novel approach to the simulation of miscible displacement by employing adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. In particular, numerical simulations of viscous fingering instabilities in heterogeneous porous media and Hele-Shaw cells are illustrated. EG is formulated by enriching the conforming continuous Gale...

We address the Korteweg-de Vries equation as an interesting model of high order partial differential equation, and show that using the classical spectral element method, i.e. a high order continuous Galerkin approximation, it is possible to develop satisfactory schemes, in terms of accuracy, computational efficiency, simplicity of implementation and, if required, conservation of the lower invar...

In this paper, space-time discontinuous Galerkin finite element method for distributed optimal control problems governed by unsteady diffusion-convectionreaction equation without control constraints is studied. Time discretization is performed by discontinuous Galerkin method with piecewise constant and linear polynomials, while symmetric interior penalty Galerkin with upwinding is used for spa...

We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretiz...

In this paper, we study the existence, regularity, and approximation of solution for a class nonlinear fractional differential equations. order to do this, suitable variational formulations are defined boundary value problems with Riemann-Liouville Caputo derivatives together homogeneous Dirichlet condition. We investigate well-posedness also regularity corresponding weak solutions. Then, devel...

The development of numerical methods for solving the Helmholtz equation, which behaves robustly with respect to the wave number, is a topic of vivid research. It was observed that the solution of the Galerkin finite element method (FEM) differs significantly from the best approximation with increasing wave number. Many attempts have been presented in the literature to eliminate this lack of rob...

We establish pointwise andW−1 ∞ estimates for finite element methods for a class of second-order quasilinear elliptic problems defined on domains Ω in Rn. These estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order. Our pointwise estimates are similar to and rely on results and analysis techniques of Schatz fo...

In this article we derive error estimates for the Galerkin approximation of a general linear second order hyperbolic partial differential equation. The results can be applied to a variety of cases e.g. vibrating systems of linked elastic bodies. The results generalize the work of Baker [1] and also allow for viscous type damping. Splitting the proofs for the semidiscrete and fully discrete case...

We consider a mixed-boundary-value/interface problem for the elliptic operator P = − ∑ ij ∂i(aij∂ju) = f on a polygonal domain Ω ⊂ R2 with straight sides. We endowed the boundary of Ω partially with Dirichlet boundary conditions u = 0 on ∂DΩ, and partially with Neumann boundary conditions ∑ ij νiaij∂ju = 0 on ∂NΩ. The coefficients aij are piecewise smooth with jump discontinuities across the in...