Wednesdays at 4:00 PM in 384H.
Nonlocality is ubiquitous in nature. While partial differential equations (PDE) have been used as effective models of many physical processes, nonlocal balanced laws are also attracting more and more attentions as alternatives to model singularities and anomalous behavior. In this talk, we discuss the mathematical structure of some nonlocal models by exploiting a recently developed nonlocal vector calculus that offers an analogy of classical vector calculus for local PDEs. The latter, when physically valid, may be derived as local limits of their nonlocal counterparts. We also present asymptotically compatible discretizations of nonlocal models that can provide convergent approximations to the local limit. Such discretizations can be more robust for multiscale problems with varying length scales.
High frequency wave propagation has been a longstanding challenge in scientific computing. For the time-harmonic problems, integral formulations and/or efficient numerical discretization often lead to dense linear systems. Such linear systems are extremely difficult to solve for standard iterative methods since they are highly indefinite. In this talk, we consider several such examples. For each one, we construct a sparsifying preconditioner that reduces the dense linear system to a sparse one and solves the problem within a small number of iterations.
The need for artificial computational boundaries in the solution of exterior wave problems, called “absorbing boundaries” among other names, arises quite often in various fields of application. In solid-earth geophysics they are needed for practically every simulation. The problem of devising good absorbing boundaries may seem very easy: we simply want to do nothing at the boundary, and let waves leave the computational domain. However, this is actually a difficult problem that has occupied researchers for many years.
During the last two decades, two classes of methods have emerged as especially powerful: the Perfectly Matched Layer (PML) method and the method of using high-order Absorbing Boundary Conditions (ABCs), which are local and involve no high derivatives. The use of ABCs has been very popular since the early 1970's, but the term “high-order ABCs” relates to the ability to implement local ABCs of an arbitrarily high order.
Recently, a new technique, called the Double Absorbing Boundary (DAB) method, has been devised for solving wave problems in unbounded domains. It has common features to high-order ABC and to PML. However, it is different from both and enjoys relative advantages with respect to both. The DAB method is first introduced in general terms, and then it is applied to the scalar wave equation in a wave guide, and to elastodynamics problems in homogeneous and heterogeneous media. Stability issues, which are especially delicate in the case of elastodynamics, are discussed in some detail.
Ongoing work and remaining challenges will be described, and in particular some problems concerning anisotropic elasticity.
For questions, contact