10:40
MS48: Mathematical and numerical solution of PDEs on manifolds (Part 2)
Chair: Elena Bachini
10:40
25 mins
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Well-balanced finite volume methods for hydrostatic equilibria on a spherical shell.
Luc Grosheintz, Roger Käppeli, Siddhartha Mishra
Abstract: A spherical shell is a frequently used a computational domain in fluid-dynamic simulations. Applications are simplified weather and climate simulations on earth or other planetary bodies, both in- and outside of our solar system.
In this talk I will present a novel, second order, three-dimensional finite volume scheme for simulating fluids on thin spherical shells. Our scheme employs layers of an icosahedral grid, which is a nearly uniform, but unstructured grid composed of hexa- and pentagons.
In sufficiently thin spherical shells a naive approximation of the geometry by flat cells, leads to the cell-center of the discrete cell to lie outside the spherical cell. We've developed a quadrature rule for the flux integral which captures the spherical nature of the cell and obeys a discrete continuity equation.
The previously mentioned examples all involve stratified fluids. This observation has two implications. The first is that the radial gradients of the density and energy are large compared to the horizontal gradients. The second is that there exists a delicate balance of the radial pressure gradient and the gravity.
We have developed a modification of the reconstruction and gravitational source term which leads to a well-balanced scheme for hydrostatic equilibria. Additionally we show that this reconstruction exactly preserves any radial symmetry in the flow, despite the unstructured nature of the grid.
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11:05
25 mins
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An unfitted finite element method for the Darcy problem in a fracture network
Alexey Chernyshenko, Maxim Olshanskii
Abstract: Numerical modelling of a flow in a fractured porous medium is a standard problem in geosciences and reservoir simulation. The problem of developing an accurate and effective numerical method for a complex network of fractures constitutes a challenge. This work introduces a finite element method for the Darcy problem posed in a system of intersecting fractures represented by a set of 2D surfaces embedded in a bulk domain. The system of fractures is allowed to cut through the background mesh in an arbitrary way. Moreover, the fractures are not triangulated in the common sense and the junctions of fractures are not fitted by the mesh. For the background mesh we use an octree grid with cubic cells. The application of the trace finite element allows to treat both planar and curvilinear fractures with the same ease. To couple the flow variables at multiple fracture junctions, we extend the Hughes--Masud formulation by including penalty terms to handle interface conditions.
The work includes numerical analysis and presents some numerical exeperiments.
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11:30
25 mins
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An Intrinsic Finite Element Method for Transport on Surfaces
Elena Bachini, Clint Dawson, Matthew Farthing, Mario Putti, Corey Trahan
Abstract: From level-set based techniques [Osher and Fedkiw, 2003] to the surface finite element method [Dziuk and Elliott, 2013] and isogeometric analysis [Ded`e and Quarteroni, 2015], a host of numerical approaches for surface PDEs have been proposed over the last twenty years. These methods have proven successful in applications from fluid flow to biomedical engineering and electromagnetism. Many, like the surface finite element method, rely on an embedding of the surface in a higher dimensional space. In this work, we consider an alternative finite element approach based on a geometrically intrinsic formulation. By careful definition of the geometry and the transport operators, we are able to arrive at an approximation that is fully intrinsic to the surface. To evaluate our approach, we consider several steady and tran- sient problems involving both diffusion and advection-dominated regimes and compare its performance to established finite element techniques.
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11:55
25 mins
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Discontinuous Galerkin Scheme for Shallow Water Flows on Surfaces
Elena Bachini, Clint Dawson, Matthew Farthing, Mario Putti
Abstract: Shallow water models of geophysical flows must be adapted to
geometrical characteristics in the presence of a general bottom
topography with non-negligible slopes and curvatures, such as a
mountain landscape. We study a shallow water model defined
intrinsically on the bottom surface from a mathematical and numerical
point of view. The equations are characterized by non-autonomous flux
functions and source terms embodying only the geometric information.
We extend an existing intrinsic first order Finite Volume scheme to
2nd order via the Discontinuous Galerkin method. Our goal is to
identify fully intrinsic discrete operators that are robust, accurate
and that explicitly maintain the symmetries of the geometrical
formulation of SWE. We compare the schemes in terms of accuracy,
stability, and robustness on simple
surfaces.
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