Electrical activity in the heart is associated with the propagation of waves with localized sharp fronts and strong gradients while for a large portion of the domain little variation is shown in all dependent variables. An other difficulty is the coupling of the electrical activity in the heart with the surrounding tissues in the torso. Anisotropic mesh adaptation has proved to be a powerful strategy to improve the accuracy and efficiency of finite element methods for solution with strong directional variations as seen with electrical waves in the heart. A main issue in our problem is the unsteady nature of the electrical waves which calls for time-dependant mesh adaptation strategies. We propose an anisotropic mesh adaptation method using a priori metric-based error estimate that is efficient for electrical waves in the heart-torso. A metric is built from metric intersection of all variables over consecutive time steps. Geometries in medical applications are often obtained from medical image segmentation, in our case using a level-set method. To deal with such level-set description of the heart-torso interface, we investigate the use of body fitted and non-body fitted meshes. An appropriate modification of the metric estimator is proposed to insure the accuracy of finite element solutions on non-body fitted meshes. In our talk, the methodology will be described in details and numerical results will show the efficiency of the approach for computing electrical waves propagation using a moving 2D heart-torso geometry.

We develop a simple and efficient adaptive mesh generator for time-dependent partial differential equations with singular solutions in two dimensional spaces. The mesh generator is based on minimizing the sum of two diagonal lengths in each cell. we also add second order difference terms to obtain smoother and more orthogonal mesh. The method is successfully applied to the nonlinear heat equations with blow-up solutions. We can obtain a solution with an amplitude of $10^{15}$ at the peak and the mesh difference of $10^{-15}$ near the peak. We also discuss nonlinear heat equations whose solutions blow up at space infinity and the blow-up time is given.

Partial differential equation (PDE) methods for pricing financial derivatives are quite popular due to their global character, ability to approximate hedging parameters, such as delta and gamma, and efficiency, at least for reasonably low dimension problems. However, various techniques that are highly associated with the numerical solution of traditional PDEs, such as adaptive mesh techniques and high-order methods, have not been widely used yet for financial PDE problems.

We consider certain financial derivative pricing problems such as valuing American options and options with jump-diffusion. We develop adaptive and high-order methods for such problems using a PDE approach. The fundamental underlying PDE for all these problems is the well-known Black-Scholes PDE, with each problem also having additional special properties. The American option pricing problem is associated with a partial differential inequality and a free-boundary problem, which is handled by a penalty iterative method. Pricing options with jump-diffusion in the asset price model involves a partial integro-differential equation (PIDE), which is handled by a fixed-point iteration.

Both finite differences and finite elements are considered for the space discretization, while classical finite differences, such as Crank-Nicolson, are used for the time discretization. The high-order discretization in space is based on an optimal finite element collocation method, the main computational requirements of which are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth-order. To control the space error, we use adaptive gridpoint distribution based on an error equidistribution principle. A time stepsize selector is used to further increase the efficiency of the methods.

Numerical examples show that our methods converge fast and provide highly accurate options prices, Greeks, and early exercise boundaries.

The Parabolic Monge-Ampere Moving Mesh Method combines equidistribution with optimal transport. We demonstrate that this method can produce an anisotropic mesh along a given feature by equidistributing a suitably chosen scalar monitor function. We define the general metric $M$, in physical space, that a mesh generated by this method aligns to. We then derive expressions for the eigenvalues and eigenvectors of $M$ for a linear feature and a feature with curvature that is radially symmetric. The eigenvectors of $M$ are shown to be orthogonal and tangential to the feature, and the ratio of the eigenvalues is shown to depend on the value of the scalar monitor density function, both locally and globally. Numerical results will be presented in 2D to verify these results.