We report our recent efforts in developing adaptive finite element methods for solving electromagnetic problems. We shall first introduce the method for the Maxwell cavity problem with discontinuous coefficients. We then consider the eddy current model with voltage excitations for complicated three dimensional structures. Unique continuation on analytic curve and its application to Inverse Prob...

The transmission eigenvalue problem plays a critical role in the theory of qualitative methods for inhomogeneous media in inverse scattering theory. Efficient computational tools for transmission eigenvalues are needed to motivate improvements to theory, and, more importantly as part of inverse algorithms for estimating material properties. In this paper, we propose two finite element methods t...

In this article, we review some of our previous work that considers the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity a...

We discuss the numerical integration of polynomials times exponential weighting functions arising from multiscale finite element computations. The new rules are more accurate than standard quadratures and are better suited to existing codes than formulas computed by symbolic integration. We test our approach in a multiscale finite element method for the 2D reaction-diffusion equation. Standard ...

In these lecture notes we introduce the finite element method and describe how it can be used to approximate the solution to certain problems of acoustic scattering. We also highlight some of the difficulties involved, and briefly summarise some current research aimed at resolving these issues.

Finite element methods for the Reissner–Mindlin plate theory are discussed. Methods in which both the tranverse displacement and the rotation are approximated by finite elements of low degree mostly suffer from locking. However a number of related methods have been devised recently which avoid locking effects. Although the finite element spaces for both the rotation and transverse displacement ...

We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous th...

Conditions for Stable Approximation Schemes Basic idea: Mimic structure of continuous problem. To establish stability of continuous problem, only used last two spaces in top sequence and last three spaces in bottom sequence. Λn−1(K) dn−1 −−−→ Λn(K)→ 0 ↗ Sn−2 ↗ Sn−1 Λn−2(V) dn−2 −−−→ Λn−1(V) dn−1 −−−→ Λn(V)→ 0. Thus, look for five finite dimensional spaces connected by a similar structure, i.e.,...

In this paper, we review the general approach to adaptivity for finite element methods presented in [l-16]. We also present new theoretical and computational results for linear elasticity, non-linear elasto-plasticity and non-linear conservation laws illustrating the general theory. The basic problem in adaptivity for finite element methods may be formulated as follows. Suppose 9 is a given (in...

Article history: Received 8 September 2012 Received in revised form 18 April 2013 Accepted 24 April 2013 Available online 22 May 2013