The mortar finite element is an example of a non-conforming method which can be used to decompose and re-compose a domain into subdomains without requiring compatibility between the meshes on the separate components. We obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used, without assuming quasiuniformity for the meshes. Our res...

Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function. Although unknowns are still associated to nodes, the function composed by piece-wise polynomials on each element and thus the gradient can be computed element-wise. Finite element spaces can thus be constructed on general triangulations and thi...

My research focuses on applied aspects of the calculus of variations and partial differential equations, particularly nonlinear equations arising from physics and chemistry, and numerical analysis and scientific computing, particularly finite-element analysis and numerical methods for interface motion. I have worked on mathematical and numerical problems arising from materials science, such as ...

In this paper, various time-domain finite-element methods for the simulation of transient electromagnetic wave phenomena are discussed. Detailed descriptions of test/trial spaces, explicit and implicit formulations, nodal and edge/facet element basis functions are given, along with the numerical stability properties of the different methods. The advantages and disadvantages of mass lumping are ...

We analyze the flux conservation property of the finite element method. It is shown that the finite element solution does approximate the flux locally in the optimal order, i.e., the same order as that of the nodal interpolation operator. We propose two methods, post-processing the finite element solutions locally. The new solutions, remaining as optimal-order solutions, are flux-conserving ele...

A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, and the condition that the shape functions depend only on the node positions. Furthermore, the method proposed also shares several of the advantages of the Finite Element Method such as: (a) the simplicity of the shape functi...

In this report, we discuss the alternating Schwarz method as the classical domain decomposition method for solving partial differential equations, which we then use to couple particle and continuum models.

We consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions. Abstract conditions are given for the multiplier spaces which are sufficient to guarantee a stable and convergent mortar approximation. Three examples of multipliers satisfying these conditions are given. The first one is a dual basis example while the remaining two are b...