For solving time-dependent convection-dominated partial differential equations (PDEs), which arise frequently in computational physics, high order numerical methods, including finite difference, finite volume, finite element and spectral methods, have been undergoing rapid developments over the past decades. In this article we give a brief survey of two selected classes of high order methods, n...

The multi-scale finite element method is used to study the vibrational characteristics of polymer matrix reinforced by single-walled silicon carbide nanotubes. For this purpose, the nanoscale finite element method is employed to simulate the nanotubes at the nanoscale. While, the polymer is considered as a continuum at the larger scale. The polymer nanotube interphase is simulated by spring ele...

We consider control-volume mixed finite element methods for the approximation of second-order elliptic problems on rectangular grids. These methods associate control volumes (covolumes) with the vector variable as well as the scalar, obtaining local algebraic representation of the vector equation (e.g., Darcy’s law) as well as the scalar equation (e.g., conservation of mass). We establish O(h2)...

Shape gradients of PDE constrained shape functionals can be stated in two equivalent ways. Both rely on the solutions of two boundary value problems (BVPs), but one involves integrating their traces on the boundary of the domain, while the other evaluates integrals in the volume. Usually, the two BVPs can only be solved approximately, for instance, by finite element methods. However, when used ...