08:30
MS48: Mathematical and numerical solution of PDEs on manifolds (Part 1)
Chair: Elena Bachini
08:30
25 mins
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Radial basis function finite differences for solving PDEs on surfaces
Grady Wright
Abstract: We discuss our recent advances in developing meshfree methods based on radial basis function generated finite differences (RBF-FD) for numerically solving biologically relevant partial differential equations (PDEs) on surfaces. The primary advantages of these methods are 1) they only require a set of nodes on the surface of interest and the corresponding normal vectors; 2) they can give high orders of accuracy; and 3) they algorithmically accessible. Commonly perceived disadvantages are that these methods require too many tuning parameters and that they are not well suited for advection-dominated problems. A goal of this talk will be to demonstrate how to overcome these issues with the use of polyharmonic spline kernels augmented with polynomials and semi-Lagrangian advection methods.
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08:55
25 mins
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Computational study of lateral phase separation in biological membranes
Sheereen Majd, Maxim Olshanskii, Annalisa Quaini, Vladimir Yushutin
Abstract: Conservative and non-conservative phase-field models are considered for the numerical simulation of lateral phase separation and coarsening in biological membranes. An unfitted finite element method is devised for these models to allow for a flexible treatment of complex shapes in the absence of an explicit surface parametrization. For a set of biologically relevant shapes and parameter values, we compare the dynamic coarsening produced by conservative and non-conservative numerical models, its dependence on certain geometric characteristics and convergence to the final equilibrium.
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09:20
25 mins
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A stable finite element method for an open, inextensible, viscoelastic rod with applications to nematode locomotion
Tom Ranner
Abstract: We present and analyse a numerical method for understanding the dynamics of an open, inextensible viscoelastic rod - a long and thin three dimensional object. Our model allows for both elastic and viscous, bending and twisting deformations and describes the evolution of the midline curve of the rod as well as an orthonormal frame which full determines the rod's three dimensional geometry. The numerical method is based on using a combination of piecewise linear and piecewise constant functions based on a novel rewriting of the model equations. We derive a stability estimate for the semi-discrete scheme and show that at the fully discrete level that we have good control over the length element and preserve the frame orthonormality conditions up to machine precision. Numerical experiments demonstrate both the good properties of the method as well as the applicability of the method for simulating locomotion of the microscopic nematode Caenorhabditis elegans.
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09:45
25 mins
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Optimal Transport Problem on surface: numerical solution via Dynamic Monge-Kantorovich
Enrico Facca, Franco Cardin, Mario Putti
Abstract: The Optimal Transport Problem (OTP) is a mathematical
problem studying minimal-cost resource reallocation. In the
recent years this problem has been studied by a number of
authors and more recently, the mathematical tools derived
from the study of the OTP have been used in more applied
fields. However, the high computational cost required for the
numerical solution of the OTP prevented a wider blending of
its ideas into applied sciences.
In (Facca et al. -SIAP 2018) we proposed a model
coupling a diffusion equation with ODE imposing a transient
dynamics to the diffusion coefficient. We conjectured
this system of equations to converge to a steady state configuration
described by the Monge-Kantorovich equations, a PDE
formulation of the OTP.
We present the theoretical and numerical evidences
supporting our claims, together with some examples of
application. The numerical scheme derived by the proposed
method results in a simple, robust, and efficient method for
the solution of the OTP on surface.
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