Applied and Computational Mathematics Seminars 2017/18

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Time: 10:00 -- 11:00

Title: CMM Center for Mathematical Modeling

Speaker: Prof. Alejandro Jofré (Director, CMM, U. de Chile)

Venue: JK Lab (room 125 Science Centre North)

Abstract:
 
In this talk I describe CMM's activities on fundamental research and applications to four main topics: Mining, optimal allocation of natural resources, BioMathematics, Bigdata in Astronomy, Marketing and mining.

 

 

Time: 14:00 -- 15:00

Title: Optimization and Energy

Speaker: Prof. Alejandro Jofré (Director, CMM, U. de Chile)

Venue: ALE 232 (room EP232 Science Centre North)

Abstract:

In this talk we introduce an optimization/game-theory model to describe an energy-producer market working on a network (transmission). We characterize a Nash equilibrium and pricing rules in this stochastic setting, as well as some stability properties.

Title:    Vortex and mass dynamics over surfaces, Maxwell laws & the axioms of mechanics

Speaker:         Stefanella Boatto
Universidade Federal do Rio de Janeiro (Brazil)
& INRIA-CetraleSupélec (France)

Date:        Wednesday, 11th October 2017

Time:        3pm

Location:     Room 1.25, O'Brien Centre for Science (North)

Abstract:
In the basic courses of mechanics a first approach to the central forces and, in particular, to the gravitational force, is made through Newton's laws and the expression of the Newtonian gravitational force

F=G m1 m2 / r^2.
It is well to remember that the 1 / r^2 dependence on the expression of force is a due contribution to Hooke's experiences in (see, among others, Arnold's book "Huygens and Barrow, Newton and Hooke"). In this approach the gravitational potential U(r) (F(x)=-\nabla_x U) is derived from the knowledge of the force.

How to find the expression of gravitational force when studying the mass dynamics in other geometries? For example on surfaces?

We have the problem of not being able to perform two-dimensional experiments to measure the force between two bodies and therefore we must find the answer to the following:

How to define a central force in an arbitrary geometry?
Given the distribution of matter on a given surface what is the fundamental equation for deducing the corresponding gravitational potential?
We propose a formulation of the dynamics directly in the intrinsic geometry of the surface and that uses fundamental solutions of the equation of the gravitational field. We show how the equations of gravitational dynamics are closely linked to those of electric charges and to the dynamics of point vortices.

Furthermore, we shall show how known laws, such as Kepler's laws and some mechanics axioms (Newton's Laws), may depend on the geometry of the space, i.e. they are not universal properties.

In collaboration with David Dritschel (Univ. of St-Andrews, Scotland), Rodrigo Schaefer (IM, UFRJ/UPC, Barcelone, Catalunya, Spain).

 

Title: Assimilating Four Decades of Observations

Speaker:   Eoin Whelan (Met Éireann)

Date:  Thursday 19th October 2017

Time:  12pm

Location:  Room 1.25, O'Brien Centre for Science (North)

Abstract:

Reanalysis is a technique used to reconstruct past climate using a state of the art Numerical Weather Prediction system.

Met Éireann has recently completed a high-resolution reanalysis of Ireland's climate, called MÉRA. This reanalysis assimilated in-situ observations over the period 1981-2015 using three-dimensional variational data assimilation  (3D-Var).

An overview of 3D-Var and validation results from MÉRA are presented.



Title:        Modeling Natural Hazards – Requirements and Selected Approaches

Speaker:     Jörn Behrens (Universität Hamburg)

Date:        Wednesday, 25th October 2017

Time:        3pm

Location:     Room 1.25, O'Brien Centre for Science (North)


Abstract:
Several extreme natural disasters in recent years - to mention the 2004 Sumatra-Andaman and 2011 Tohoku Tsunami events, 2005 Hurricane Katrina, 2013 Taifun Haiyan, 2010 Eyjafjallajökull Volcano eruption - reminded the world of the vulnerability of modern societies and highly populated areas. In order to save life and property, early warning and preparedness is essential, which in turn requires fast and accurate simulation results for forecasting and scenario simulations. This presentation discusses requirements in such operations from a mathematical and computational point of view. One major concern is the multi-scale character of many such problem settings. Solution approaches to the requirements of multi-scale representation, accuracy, efficiency and visualization are presented. We try to extract overarching computational strategies for disaster mitigation.




Title:    Overview of the Marine Institute's Ocean Modelling Activities

Speaker:         Tomasz Dabrowski (Marine Institute)

Date:        Wednesday, 1st November 2017

Time:        3pm

Location:     Room 1.25, O'Brien Centre for Science (North)

Abstract:
Operational hydrodynamic models are run daily in an operational automated system at the Marine Institute to produce both hindcasts and forecasts to build up a database of modelled data to support marine research. The operational modelling suite at the MI is based on the open-source community driven Regional Ocean Modelling System (ROMS) for ocean physics and Simulating WAves Nearshore (SWAN) and use operationally available atmospheric and open ocean boundary forcing data. The Marine Institute are also responsible for the biogeochemical modelling service within the framework of the Copernicus Marine Environment Monitoring Service. The biogeochemical model used is PISCES. Downstream modelling services include the areas of aquaculture, fisheries, oil spill modelling, search and rescue, storm surges and climate studies. The presentation will outline the set-up of the models and an will provide an overview of the downstream services.



Title:    Models of large boulder generation and transport by waves.

Speaker:     James Herterich (UCD)

Date:    Wednesday, 15th November 2017

Time:        3pm

Location:        Room 1.25, O'Brien Centre for Science (North)


Abstract:
Coastal boulder deposits consist of megagravel, with 100+ tonne blocks close to sea level, as well as boulders of many 10s of tonnes mass emplaced at elevations 10s of metres above high water and up to 250m inland. The largest boulders can be moved by several metres during storm events. Understanding the mechanisms of boulder generation and transport gives a unique insight to the powerful events that occur along high-energy coastlines.

We present a model of boulder generation via a hydraulic press-like action on exposed beams on platforms. The stress generated in the rock can propagate cracks to full fracture. Hydrodynamics, mechanics, and fracture models are discussed in relation to the problem, alongside field evidence.

The subsequent transport of large boulders by overtopping waves and surges has been actively modelled for the last 20+ years. We discuss some fundamentals in the models, and show improvements in determining the forces involved in a number of situations. We use complex variable techniques, treating the boulder as an obstacle in the flow.



Title:        Scattering of two spinning black holes in post-Minkowskian gravity.

Speaker:     Justin Vines ,Max-Planck Institute for Gravitational Physics.

Date:        Wednesday, 22nd November 2017

Time:        3pm

Location:     Room 1.25, O'Brien Centre for Science (North)


Abstract:
The advent of gravitational-wave astronomy motivates detailed study of the classical gravitational dynamics of spinning black holes in binary systems.  Analytical calculations to those ends (for arbitrary mass ratios) have traditionally focused on post-Newtonian perturbation theory (expansion in 1/c), naturally paired with an expansion in small spins.  I will discuss a complimentary approach using post-Minkowskian perturbation theory (expansion in G), paired with a nonperturbative treatment of the spins.  New results for two-spinning-black-hole scattering at first post-Minkowskian order (linear order in G), given in simple closed forms to all orders in spin, reveal remarkable relationships between arbitrary-mass-ratio two-body dynamics and test-body dynamics in a fixed background, and between the dynamics of spinning and nonspinning black holes.


Title:    Some recent developments in tsunami observations and theory

Speaker:         Emile Okal (Northwestern University)

Date:        Monday, 4th December 2017

Time:        2pm

Location:     Room 1.25, O'Brien Centre for Science (North)

Abstract:
We present three recent observational and theoretical developments in tsunami science


1.    While it has long been known that tsunamis can be focused and defocused by bathymetric heterogeneities, we consider the case of the sharp refraction expected at a continental shelf, where velocities can vary by as much as a factor of 4 to 5. We confirm that Snell's law is indeed upheld, using both real and simulated time series. This implies that in the presence of a wide continental shelf, distant tsunamis can be considered as impacting the coast at normal incidence.


2.    Motivated by our experience during the 2011 Tohoku tsunami in Tahiti, we explore the conditions under which the leading wave may or may not be that with maximum amplitude. Based on available analytical solutions for simple source models, we find that this "sequencing" of tsunami waves is due to the initiation of dispersion outside the Shallow-Water-Approximation, and as such controlled by a simple combination of distance, effective source size and water depth.


3.    Following the detection of a millimetric tsunami by DART buoys during the 2013 deep Okhotsk earthquake, we explore theoretically the excitation of tsunamis by deep earthquakes. While known events have not produced tsunamis with damaging potential, our lack of knowledge of the maximum size of deep earthquakes leaves open this possibility.


Title:    A regime-based perspective on variability of the Southern Hemisphere mid-latitude jet

Speaker:         Nick Byrne (Reading University)

Date:            Wednesday, 13th December 2017

Time:            3pm

Location:         Room 1.25, O'Brien Centre for Science (North)

Abstract:
 
Variability of the large-scale mid-latitude Southern Hemisphere circulation is dominated by apparently random wanderings of the mid-latitude jet. These wanderings of the jet have an important influence on a variety of components of the Southern Hemisphere climate system, including the location of synoptic storms and Antarctic sea-ice extent. For example, the unprecedented retreat of Antarctic sea-ice that occurred during the spring of 2016 has been linked to unusual behaviour of the jet and increased synoptic storm activity.
In this talk several lines of evidence will be presented to argue that variations of the Southern Hemisphere mid-latitude jet are more systematic than previously thought. In particular, it will be argued that jet variability during austral spring and summer is closely coupled to the seasonal cycle of the polar vortex in the stratosphere. Based on this connection with the seasonal cycle of the stratospheric circulation, an alternative perspective of tropospheric jet variability will be proposed. This alternative perspective will subsequently be used to re-interpret several previous results in the literature, and also to motivate a partial explanation for the unprecedented Antarctic sea-ice behaviour during spring 2016.

 


Title: Acoustic-gravity waves, theory & applications

Speaker: Usama Kadri (Cardiff University)

Date: Wednesday, 24 January, 2018

Time: 3pm

Location: SCN 1.25, O'Brien Centre for Science (North)


Abstract:


Acoustic–gravity waves (AGWs) are compression-type waves generated as a response to a sudden change in the water pressure, e.g. due to nonlinear interaction of surface waves, submarine earthquakes, landslides, falling meteorites and objects impacting the sea surface.

AGWs can travel at near the speed of sound in water (ca. 1500 m/s), but can also penetrate through the sea-floor surface amplifying their speed, which turns them into excellent precursors. “Acoustic–gravity waves” is an emerging field that is rapidly gaining popularity among the scientific community, as it finds broad utility in physical oceanography, marine biology, geophysics, water engineering, and quantum analogues.

This talk is an overview on AGWs, with emphasis on recent developments, current challenges, and future directions.

 Title:    From the wave spectrum to power output: statistical aspects and efficient computational tools

Speaker:      Alexis Merigaud (Maynooth)

Date:        Wednesday, 31st January 2018

Time:         3pm

Location:        Room 1.25 O’Brien Centre for Science North

Abstract:

This presentation will consist of two short parts:

1)    Most of the time in the power production regime of wave energy converters (WECs), the wave elevation can be described as a stationary, homogeneous Gaussian random field, if the temporal and spatial areas considered are "small enough" (~30 min, 1-2 km). In the light of the theory of discrete-time stochastic processes, we briefly discuss possible statistical representations of the wave elevation, when sampled at regular time intervals through a measurement system. We find that auto-regressive processes or the (more general) class of regular processes are two appropriate and useful descriptions. We then show how those two characterisations lead to short-term wave elevation prediction techniques.

2)    Computationally-efficient simulation methods are useful for WEC optimisation or power assessment. Relying on a projection of the system variables and equations onto a Fourier basis, the harmonic balance (HB) method can be used to compute the steady-state response of a non-linear system subject to a periodic input signal. Applications of the technique to non-linear WEC simulation in irregular waves are shown.


Three short (~12 min.) presentations by the PhD students in the Weather, Climate and Energy research group.



Date:          Wednesday 14th February 2018

Time:           3pm

Location:        Room 1.25, Science Centre North



Talk 1

Title: Shortwave Radiation in Reanalyses: Skill Scores and Spatial Patterns.

by: Eadaoin Doddy

I will present work done to assess the accuracy of different reanalyses for daily shortwave radiation (SW) values across Ireland. The high-resolution regional reanalysis, MÉRA and two low-resolution global reanalyses; ERA-Interim and MERRA2 are compared to observations for up to 30 years. Post-processing methods are explored with a view to providing an improved dataset for the renewable energy community in Ireland. The spatial pattern of SW is also assessed for the three reanalysis datasets. Daily SW varies in an east-west direction which is highlighted by the land-sea contrast. MÉRA does well in capturing this spatial pattern. ERA-Interim also captures this pattern, however it is less pronounced in MERRA2. Satellite data will be used to explore reasons for this SW spatial pattern. I will end with a discussion of future work involving adaptive spatial multivariate post-processing for the renewable energy sector in Ireland.



Talk 2

Title: Wind-Solar Correlations in Reanalysis Datasets

by: Seánie Griffin

Combining wind and solar power production has the potential to reduce the overall variability of renewable energy generation. Skill scores for 10m wind speed are assessed for MÉRA, the high-resolution reanalysis produced by Met Éireann, and also two coarse resolution global reanalyses: ERA-Interim and MERRA2. Wind-solar correlations are compared to those calculated using observed data from stations around Ireland, across different timescales. Reanalysis datasets are found to frequently overestimate the strength of the negative correlation between wind speed and shortwave radiation. Correlations are also seen to vary with wind direction. Finally an overview will be given of future work to improve the skill of Numerical Weather Prediction models for renewable energy forecasting.



Talk 3

Title: Large scale atmospheric pressure patterns and wind and solar season to season variability – a case study for Ireland and the UK

by: Joao Monteiro Correia

In Ireland and the UK, long-term atmospheric variability has been linked previously with large-scale atmospheric patterns such as the North Atlantic Oscillation (NAO), the East Atlantic (EA) and Scandinavian (SCAND) patterns. These patterns, identified from pressure anomalies, influence other meteorological variables relevant to renewable power generation. Further assessments of these influences are required for successful integration of renewable energy technologies in the energy grid. Using various datasets, this work explores the links between the atmospheric patterns referred to above and both wind speed and solar radiation in the region of interest. The main result of my research so far is that winter solar radiation variability is linked strongly to the NAO and SCAND patterns, and that that relationship expresses itself in zonal gradients, across both Ireland and the UK.


Title:    Towards improved phenomenological waveform models
 
Speaker:         Marta Colleoni (University of the Balearic Islands)

Date:        Wednesday, 28th February 2018

Time:        3pm

Location:     Room 1.25, Science Centre North


Abstract:
In preparation for the new Advanced LIGO Observation Run (O3), the waveform models employed by gravitational wave searches are being refined in order to achieve a better recovery of the parameters of the sources. The so-called Phenom(enological) templates represent a flexible and cost-effective tool to study the signals emitted during compact binary coalescences. I will describe how Phenom models are currently being extended to improve their coverage of the parameter space and increase detection efficiency.



Title:    A Brief History of Gravitational Wave Emission

Speaker:     Daniel Kennefick (University of Arkansas)

Date:    Wednesday, 7th March 2018

Time:    3pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
Over a billion years ago gravitational waves from two colliding black holes started on their journey towards Earth.  A century ago Albert Einstein predicted their existence.  Then he changed his mind.  Then he changed his mind again.  Others joined the argument.  Finally, forty years ago evidence that they were real came when a binary neutron star was discovered in our galaxy which was decaying in our orbit in just the fashion Einstein predicted, having emitted gravitational waves 21,000 years ago. Subsequently large detectors were constructed on the Earth, such as LIGO, GEO600 and VIRGO to try to detect gravitational waves.  But so difficult was theoretically modelling what they would look like that twenty years ago recent Nobel laureate Kip Thorne made a wager that theorists would still not have predicted their form before the detectors saw them.  In 2005 a breakthrough was made in supercomputer simulations of binary black holes so that templates were available just in time when the gravitational waves arrived on Earth two years ago after their billion year journey.


Title:    Southern Hemisphere summertime Rossby waves and weather in the Australian region

Speaker:         Laura Cooke (UCD)

Date:            Wednesday, 28th March 2018

Time:            3pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
Motivated by recent studies on significant weather patterns in Australia, in this talk I will investigate the life cycle of summertime transient Rossby wave packets in the Southern Hemisphere.

A variable, I, representing the strength of the interaction between wave packets and the jet stream is used as the basis for composites in 5 regions of the Southern Hemisphere. In each case, cyclogenesis precedes a very clear transient Rossby wave packet. While in most cases, waves propagate far downstream, in some they also refract strongly equatorward and are associated with enhanced tropical convection. I will discuss the characteristics of the different wave behaviours and their associated weather patterns.

In addition, I will summarise my new role at UCD and proposed research that I will undertake with Conor Sweeney's group through the Energy Systems Integration partnership Programme (ESIPP).
 Figure caption: DJF seasonal average of I: Advection of potential vorticity by the divergent part of the wind (PVU s−1 ) at 350 K (shaded) with boxed regions (pink) and the 350 K jet isotachs [20, 25, 30, 35] ms−1 (green contours)





Title:    A Posteriori Error Estimation and Adaptive Algorithms for Computational Fluid Dynamics

Speaker:         Professor Johan Hoffman (KTH, Sweden)

Date:            Thursday 5th April 2018

Time:            1pm        (Please note non-standard time and date)

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
We present recent developments on a posteriori error estimation and adaptive algorithms for Computational Fluid Dynamics (CFD), with particular emphasis on turbulent flow, complex geometry, coupled problems and high performance computing. Turbulent flow poses challenges both with regards to the cost of resolving fine scale turbulent structures, in the bulk of the flow as well as near solid boundaries, and with regards to predictability of quantities of interest as turbulent flows show chaotic features. We review of our work towards a parameter-free method for simulation of turbulent flow at high Reynolds numbers, where we develop a model for turbulent flow in the form of weak solutions of the Navier–Stokes equations, approximated by an adaptive finite element method, where: (i) viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and (ii) skin friction at solid walls is assumed to be negligible compared to inertial effects. The result is a computational model without empirical data, where the only model parameter is the local size of the finite element mesh. Under adaptive refinement of the mesh based on a posteriori error estimation, output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution, and thus the resulting method has no adjustable parameters. No ad hoc design of the mesh is needed, instead the mesh is optimized based on solution features, in particular no boundary layer mesh is needed. We connect the computational method to the mathematical concept of a dissipative weak solution of the Euler equations, as a model of high Reynolds number turbulent flow, and we highlight a number of benchmark problems for which the method is validated.

Tea, Coffee and Light Refreshments: School of Mathematics and Statistics Common Room at 12.20






Title:    Subspace augmentation for iterative methods for well- and ill-posed problems: recent results and insights


Speaker:     Kirk Soodhalter (TCD)


Date:    Wednesday 11th April 2018


Time:    3pm


Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
Augmented Krylov subspace methods are a family of techniques which have been proposed in the context of solving both well- and ill-posed linear problems, at times with mathematically similar methods often being proposed in different communities. However, the goals when using such methods in those two contexts differs. In any Krylov subspace method, one iteratively generates a subspace from which solution approximations are drawn. In an augmentation method, one seeks to somehow enrich this space with additional information. For well-posed problems, this done both to damp the influence of parts of the spectrum which can slow convergence and to mitigate the effects of „restarting“, wherein one must discard the generated subspace due to memory constraints. For ill-posed problems (in image and signal reconstruction), these methods have been proposed for the improvement of the reconstruction and acceleration of the semiconvergence, particularly in the case where one augments with known sharp edge (i.e., large gradient-norm) features and jumps.

In this talk, we place all these methods into a common framework, which allows us to more easily relate them to one another. One can then use this understanding to standardize the process of combining the augmentation strategy with other existing iterative methods. This is demonstrated with, e.g., the Arnoldi-Tikhonov method. Numerical examples from the well- and ill-posed problems communities will be presented.





 

Title:    HPC for Computational Sciences: Landscape and Highways

Speaker:         Venkatesh Kannan
(Irish Centre for High-End Computing)

Date:            Wednesday 18th April 2018

Time:            3pm

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
High performance computing (HPC) has for long been an integral part of computational modelling and simulation. Recently, the HPC landscape is evolving rapidly, two aspects of which will be addressed in this talk: (1) an overview of the computing devices that will benefit the scientific community at large, and (2) pointers to modern programming methods, libraries and tools for faster development of efficient HPC code. The conclusion will include an overview of ICHEC and its activities.




Title:    Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

Speaker:      Professor John D. Gibbon
(Imperial College London)

Date:        Thursday 19th April 2018

Time:        1pm        (Please note unusual time and day)

Location:     Room 1.25, O’Brien Centre for Science (North)


Abstract:
Whether the 3D Navier-Stokes equations have regular solutions for arbitrary long times has been an enduring problem since the time when Jean Leray (1934) set out his ideas on weak solutions. After more than a century, conventional methods appear to have been exhausted. My talk will summarize a set of numerical experiments that suggest that nonlinear depletion in 3D-NS dynamics is very strong. In this context, solutions appear to operate in a regular regime. It is shown that two other regimes theoretically exist, one of which corresponds to weak solutions but neither of these appear to be represented in the dynamics. However, a transition between the three cannot be discounted.


Title:        The Rayleigh-Taylor instability

Speaker:     John Gibbon  -  (Imperial College London)

Date:        Wednesday 25th April 2018

Time:        3pm

Location:     Room 1.25, O’Brien Centre for Science (North)


Abstract:
The Rayleigh-Taylor instability (RTI) is a phenomenon with a long history. It appears in laboratory mixing experiments, laser plasma devices and astro-physical phenomena. The RTI in miscible fluids can be modelled by considering the evolution of the gradient of the composition density in a buoyancy-driven turbulent flow. Data first produced in simulations by Livescu & Ristorcelli (2007) (available on the Johns Hopkins Turbulence Database) is analysed in terms of this model which shows extremely rapid growth in the density gradient. In contrast, the RTI also appears in immiscible fluids, particularly in materials science where phase field physicists use the Cahn-Hilliard-Navier-Stokes equations to model this phenomenon. I will present some brief mathematical results on this latter problem.


Title:    Probing Fundamental Bounds in Hydrodynamics Using Variational Optimization Methods

Speaker:         Bartosz Protas  -  (McMaster University)

Date:                                                  Thursday 26th April 2018

Time:        1pm        (Please note unusual day and time)

Location:     Room 1.25, O’Brien Centre for Science (North)

Abstract:
In the presentation we will discuss our research program concerning the study of extreme vortex events in viscous incompressible flows. These vortex states arise as the flows saturating certain fundamental mathematical estimates, such as the bounds on the maximum enstrophy growth in 3D (Lu & Doering, 2008). They are therefore intimately related to the question of singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it allows one to determine whether or not certain mathematical estimates are "sharp", in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. In the presentation we will review a number of results concerning 2D and 3D flows characterized by the maximum possible growth of, respectively, palinstrophy and enstrophy. It will be shown that certain types of initial data, such as the Taylor-Green vortex, which have been used in numerous computational studies of the blow-up problem are in fact a particular instance (corresponding to an asymptotic limit) of our family of extreme vortex states. We will present results comparing the growth of relevant quantities in high-resolution direct numerical simulations of the Navier-Stokes system obtained using our extreme vortex states and different initial data employed in earlier studies. Since none of the 3D computations reveals any tendency for the enstrophy to become unbounded in finite time, the main conclusion is that, should finite-time blow-up be indeed possible in the 3D Navier-Stokes system, it is unlikely to arise from initial data maximizing the instantaneous growth of enstrophy
[Joint work with Diego Ayala and Dongfang Yun]

Title:             Bivariate Splines for Numerical Solution of Helmholtz Equation with Large Wave Number       
         
Speakers:    Dr. Ming-Jun Lai (University of Georgia)

Date:           Thursday 10th May

Time:           2pm

Location:    Room A005, Health Sciences Building

Abstract:


We use bivariate splines to solve Helmholtz equation with large wave number, e.g. wave number k=1000 or larger while the size of underlying triangulation is reasonable. It is known that for linear finite elements, the wave number kand the size h of triangulation satisfies a stubborn relation k2h<1. For high order finite elements, so-called hp version of internal penalty discontinuous Galerkin (IPDG) methods, the relation is k3h2=O(p2) for polynomial degree p. For continuous internal penalty finite element method (CIP-FEM), the relation is O(k2p+1h2p)=O(k1+1/(2p)hp). That is, when k=1000, h has to be extremely small, i.e. the size of underlying triangulation has to be extremely small which leads to the infeasibility of numerical computation. In this talk, we demonstrate that the bivariate spline method enables us to have very accurate solution for kh ≥ p/2 when we use p ≥ 10. No pollution phenomenon is observed in our experiments as long as kh=p/2. We then extend our spline method to numerical solution of a Helmholtz problem over inhomogeneous media and a wave equation with periodic source arising from Maxwell's equations in potential formulation. In addition, I will explain the existence, uniqueness and stability of our spline solutions as well as their approximation.