In 1996 Hazard and Lenoir suggested a variational formulation of Maxwell's equations using an overlapping integral equation and volume representation of the solution [SIAM J. Math. Anal., 27 (1996), pp. 1597--1630]. They suggested a numerical scheme based on this approach, but no error analysis was provided. In this paper, we provide a convergence analysis of an edge finite element scheme for the method. The analysis uses the theory of collectively compact operators. Its novelty is that a perturbation argument is needed to obtain error estimates for the solution of the discrete problem that is best suited for implementation.

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility of these algorithms, we present representative calculations for reduced free rigid body motion (a conservative ODE) and a discretization of micromagnetics (a dissipative PDE). We emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms; this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions.

Sharp upper bounds for the first eigenvalue of the Laplacian on a surface of a fixed area are known only in genera zero and one. We investigate the genus two case and conjecture that the first eigenvalue is maximized on a singular surface which is realized as a double branched covering over a sphere. The six ramification points are chosen in such a way that this surface has a complex structure of the Bolza surface. We prove that our conjecture follows from a lower bound on the first eigenvalue of a certain mixed Dirichlet-Neumann boundary value problem on a half-disk. The latter can be studied numerically, and we present conclusive evidence supporting the conjecture.

We present a boundary integral method, and an accompanying boundary element discretization, for solving boundary-value problems for the Laplace-Beltrami operator on the surface of the unit sphere S in R3. We consider a closed curve C on S which divides S into two parts S1 and S2. In particular, C = ∂ S1 is the boundary curve of S1. We are interested in solving a boundary value problem for the Laplace-Beltrami operator in S2, with boundary data prescribed on C.

We examine the effect of numerical integration on the convergence of high order pyramidal finite element methods. Rational functions are indispensable to the construction of pyramidal interpolants so the conventional treatment of numerical integration, which requires that the finite element approximation space is piecewise polynomial, cannot be applied. We develop an analysis that allows the finite element approximation space to include rational functions and show that despite this complication, conventional rules of thumb can still be used to select appropriate quadrature methods on pyramids. Along the way, we present a new family of high order pyramidal finite elements for each of the spaces of the de Rham complex.

We present a family of high-order finite element approximation spaces on a pyramid, and associated unisolvent degrees of freedom. These spaces consist of rational basis functions. We establish conforming, exactness and polynomial approximation properties.

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d ≥ 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen-Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that a series of published numerical studies are problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit.

We present and analyze a multigrid algorithm for the acoustic single layer equation in two dimensions. The boundary element formulation of the equation is based on piecewise constant test functions and we make use of a weak inner product in the multigrid scheme as proposed in [BLP94]. A full error analysis of the algorithm is presented. We also conduct a numerical study of the effect of the weak inner product on the oscillatory behavior of the eigenfunctions for the Laplace single layer operator.

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere in the presence of multiple "islands" are presented. The surface of the sphere is first mapped to a multiply-connected region in the complex plane via a stereographic projection. After discretizing the integral equation, the resulting dense linear system is solved iteratively using the fast multipole method for the 2D Coulomb potential in order to calculate the matrix-vector products. This numerical scheme requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. The performance of the method is demonstrated on several examples.

Skeletal muscle can contain neuromuscular compartments that are spatially distinct regions that can receive relatively independent levels of activation. This study tested how the magnitude and direction of the force developed by a whole muscle would change when the muscle activity was regionalized within the muscle. A 3D finite element model of a muscle with its bounding aponeurosis was developed for the lateral gastrocnemius, and isometric contractions were simulated for a series of conditions with either a uniform activation pattern, or regionally distinct activation patterns: in all cases the mean activation from all fibers within the muscle reached 10%. The models showed emergent features of the fiber geometry that matched physiological characteristics: with fibers shortening, rotating to greater pennation, adopting curved trajectories in 3D and changes in the thickness and width of the muscle belly. Simulations were repeated for muscle with compliant, normal and stiff aponeurosis and the aponeurosis stiffness affected the changes to the fiber geometry and the resultant muscle force. Changing the regionalization of the activity resulted to changes in the magnitude, direction and center of the force vector from the whole muscle. Regionalizing the muscle activity resulted in greater muscle force than the simulation with uniform activity across the muscle belly. The study shows how the force from a muscle depends on the complex interactions between the muscle fibers and connective tissues and the region of muscle that is active.

In this dissertation, we consider a class of boundary value problems which are posed exterior to a thin domain. We are interested in the asymptotic behavior of the solutions as the thickness of the thin region approaches zero.

A general solution procedure for the class under consideration is proposed. The computational domain is reduced to a finite region by means of integral equations, and a variational formulation for the resulting nonlocal boundary value problem is carefully studied. Certain a priori estimates are established. Thereafter, we scale the domain along the thickness, and use a regular asymptotic expansion to approximate the exact solution. This results in a family of simpler variational problems to be solved. We examine solvability issues for this sequence of problems, and may use the a priori estimates to justify the procedure.

The study is carried out with reference to three specific problems which belong to this class. We first exhaustively study a simple model problem, and apply the techniques developed to subsequently analyse a time-harmonic scattering problem. In both these cases we are able to rigorously justify the asymptotic procedure. We then consider a fluid-structure interaction problem, and extend the general procedure to study it. Finally, some computational results are presented.