This paper proposes semi-discrete and fully discrete hybridizable discontinuous Galerkin (HDG) methods for the Burgers' equation in two three dimensions. In spatial discretization, we use piecewise polynomials of degrees $ k \ (k \geq 1), k-1 l (l = k-1; k) to approximate scalar function, flux variable interface trace respectively. full discretization method, apply a backward Euler scheme tempo...

An approach to solve nite time horizon sub-optimal feedback control problems for partial di erential equations is proposed by solving dynamic programming equations on adaptive sparse grids. The approach is illustrated for the wave equation and an extension to equations of Schrödinger type is indicated. A semi-discrete optimal control problem is introduced and the feedback control is derived fro...

We present a mixed finite element method for a model of the flow in a Hele-Shaw cell of 2-D fluid droplets surrounded by air driven by surface tension and actuated by an electric field. The application of interest regards a micro-fluidic device called ElectroWetting on Dielectric (EWOD). Our analysis first focuses on the time-discrete (continuous in space) problem and is presented in a mixed va...

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties–in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes th...

Rheological behavior of certain non–Newtonian fluids in engineering sciences is often modeled by power law ansatzes with p ≤ 2. So far, existing numerical analysis for local strong solutions study a fully implicit time discretization and find only restricted ranges of admissible p’s for corresponding error estimates [9]; different nonlinear stabilization strategies which allow a corresponding e...

Magnetic Resonance Imaging (MRI) is a widely applied non-invasive imaging modality based on non-ionizing radiation which gives excellent images and soft tissue contrast of living tissues. We consider the modified Bloch problem as a model of MRI for flowing spins in an incompressible flow field. After establishing the well-posedness of the corresponding evolution problem, we analyze its spatial ...

We present preliminary results of the numerical simulation of electrocardiograms (ECG). We consider the bidomain equations to model the electrical activity of the heart and a Laplace equation for the torso. The ionic activity is modeled with a Mitchell-Schaeffer dynamics. We use adaptive semi-implicit BDF schemes for the time discretization and a Neumann-Robin domain decomposition algorithm for...

For the unsteady convection-diiusion equation in two dimensions we derive a new cell-based semi-discretization which is founded on the method of lines and a nite volume approach. Moreover, we present a second semidiscretization technique, a nonconforming Petrov-Galerkin method with exponentially tted trial and test functions. If we use appropriate quadrature rules both approaches are equivalent...

A semi-discretization method for solving an age-dependent population dynamics model with an additional structure is proposed. This method, unlike previous ones, considers the partial discretization which reduces the model equation into a first order ordinary differential equation. The latter is then solved explicitly and conditions under which second order accuracy arises are given. While the a...

We study anisotropic surface diffusion of curves with a small corner energy regularization. The regularization allows the use of nonconvex free energy densities and turns the evolution law into a 6th order geometric equation. Using a semi-implicit time discretization, we present a variational formulation of this equation for parametric curves, leading to a discretization based on linear finite ...