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In this work, several nonlocal problems are studied. Analysis and computation have been done for these problems. Firstly, we consider the time-dependent nonlocal diffusion and wave equations, formulated in the peridynamics setting....
Overcoming Geometric Limitations in the Finite Element Method by Means of Polynomial Extension: Application to Second Order Elliptic Boundary Value and Interface Problems
In this dissertation, we present a new approach for approximating the solution of second order partial differential equations and interface problems. This approach is based on the classical finite element method, where instead of using...
The objective of this work is the development of novel, efficient and reliable sparse grid stochastic collocation methods for solving linear and nonlinear partial differential equations (PDEs) with random coefficients and forcing terms ...
Mass Conserving Hamiltonian-Structure-Preserving Reduced Order Modeling for the Rotating Shallow Water Equations Discretized by a Mimetic Spatial Scheme
Ocean modeling, in a climate-modeling context, requires long time-horizons over global scales, which when combined with accurate resolution in time and space makes simulations very time-consuming. While high-resolution ocean-modeling...
This work focuses on developing and analyzing novel, efficient sparse-grid algorithms for solving several types of stochastic ordinary/partial differential equations and corresponding inverse problem, such as parameter identification....
The nonlocal models considered are free of spatial derivatives and thus are suitable for modeling problems with solutions exhibiting defects such as fractures in solids. Those models feature a horizon parameter that specifies the maximum...
Stochastic Navier-Stokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and...
Mesh generation in regions in Euclidean space is a central task in computational science, especially for commonly used numerical methods for the solution of partial differential equations (PDEs), e.g., finite element and finite volume...
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