University of Texas at Austin

Cross-
Cutting
Research Area

Numerical Analysis and Numerical PDEs

The study of algorithms for numerical approximation

Development, analysis, and computer implementation for numerical simulation of mathematical models

Advanced mathematical analysis can lead to stable, accurate, and efficient computer algorithms that faithfully represent important features of the system being approximated. These include preserving physical principles like conservation of mass and energy, capturing phenomena on multiple spatial and temporal scales, and accounting for uncertainties in the data and even in the model itself.

An Overview: Numerical Analysis & Numerical PDEs

What are Numerical Analysis & Numerical PDEs

Numerical Analysis is a branch of mathematics and computer science that creates, analyzes, and implements algorithms that use numerical approximation. The recent growth in computing power has revolutionized the use of mathematical modeling, and at the same time it revealed subtle and sometimes mysterious numerical issues. Questions addressed, for example, may concern accuracy and error estimation, stability of the model to small perturbations including rounding error, preservation of physical principles, multiscale modeling and model adaptivity, data assimilation and inverse modeling, predictive science, computational complexity, machine learning, computational efficiency, software, and computer hardware.

Numerical PDEs is the subset of numerical analysis concerned with the numerical approximation of partial differential equations (PDEs), integro-differential equations (IDEs), and optimization and control. Systems of PDEs and IDEs with inequality constraints underlie many if not most of the mathematical models arising in the natural and social sciences, engineering, medicine, and business.

Current research areas

Research is multifaceted, ranging from foundational advances in theory, methods and algorithms, to real-world impact in grand challenge problems. Some examples of current research areas include:

Finite element method

Finite element methods. The aim is to solve all kinds of PDEs efficiently and to within specified error bounds using unstructured, polyhedral, and refined computational meshes.

Discontinuous Galerkin

Discontinuous Galerkin (DG) and Discontinuous Petrov Galerkin (DPG) methods. These are special finite element methods, and the goal is to understand and exploit their many advantageous properties.

isometric analysis

Isogeometric analysis. The vision is an integration of computational geometry and analysis, by unifying the disparate methods and data structures of Computer Aided Geometric Design (CAGD) and finite element analysis.

Approximation of div-curl PDE systems

Approximation of div-curl PDE systems. These arise in many problems of mechanics and physics, such as those involving electromagnetism, acoustics, and diffusion. They have a precise mathematical structure, and the challenge is to preserve this structure within the numerical methods.

Fluid structure interaction (FSI) problems

Fluid structure interaction (FSI) problems. These arise in many important applications, and involve complex numerical issues related to mechanical structures deforming within a fluid.

Hyperbolic conservation laws

Hyperbolic conservation laws. These PDEs model, e.g., transport processes, fluid dynamics, and turbulence. The equations offer special challenges related to the interaction of numerics and physics, such as arise in the formation of shocks and dealing with nonuniqueness of solutions.

Boltzmann type equations and particle transport

Boltzmann type equations and particle transport. Challenges include preservation of physics in numerical methods at quantum, kinetic, and fluid scales, and approximation of high-dimensional non-equilibrium statistical flows.

Multiphysics simulation

Multiphysics simulation. A complex system often contains many smaller, coupled subsystems. Solving them iteratively leads to possible nonconvergence of the overall problem and issues of propagation of errors between subsystems.

Solution of linear systems

Solution of linear systems. Most numerical methods reduce at some level to very large systems of linear equations, and the overall computational cost is related most strongly to the time needed to solve these linear systems. Algorithms using direct, probabilistic, and iterative methods are explored in a massively parallel, high performance computational setting.

Working with partners

Centers and Groups

To learn more about projects and people in Numerical Analysis and Numerical PDEs, explore the centers and groups with research activities in this cross-cutting research area.

Applied Mathematics Group

Center for Numerical Analysis

Computational Mechanics Group

News in brief

Kate Pearce Awarded NSF-DMS Ascend Postdoctoral Fellowship

News

Oct. 17, 2024

Kate Pearce Awarded NSF-DMS Ascend Postdoctoral Fellowship

Postdoctoral Researcher and prison education advocate Kate Pearce was awarded the NSF-DMS Ascend Postdoctoral Fellowship to support her work on rank-structured matrix compression and building mathematics course offerings at Lockhart Women's Correctional Facility. 

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Renowned Pioneer in Computational Sciences Mary Wheeler Retires

Feature

Oct. 7, 2024

Renowned Pioneer in Computational Sciences Mary Wheeler Retires

Mary F. Wheeler, best known for her work in finite element analysis and porous media problems, retired in August 2024.

Her research uses an interdisciplinary approach to solve grand challenges with applications from environmental remediation, carbon storage, thermal energy to ideas in predictive surgery.

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Kun Huang Wins Oden Institute 2024 Outstanding Dissertation Award

News

March 27, 2024

Kun Huang Wins Oden Institute 2024 Outstanding Dissertation Award

Kun Huang's dissertation focuses on advancing numerical methods for solving the kinetic equation in plasma physics. 

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Bringing it all Together - Profile Jonathan Zhang

Profile

Feb. 21, 2024

Bringing it all Together - Profile Jonathan Zhang

Jonathan Zhang, a third-year CSEM Graduate Student, explores his journey to the Oden Institute and the impact of community on his experience.

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Etching Progress into Memory: Leszek Demkowicz’ 'Mathematical Theory of Finite Elements’ Published by SIAM

News

Jan. 9, 2024

Etching Progress into Memory: Leszek Demkowicz’ 'Mathematical Theory of Finite Elements’ Published by SIAM

The Mathematical Theory of Finite Elements, a work that began as a collection of lecture notes, has developed into a powerful addition to Core Faculty Member Leszek Demkowicz's collection of published textbooks. 

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International Workshop Held for Leszek Demkowicz’s 70th Birthday

News

April 24, 2023

International Workshop Held for Leszek Demkowicz’s 70th Birthday

Guests from around the globe gathered for a celebratory workshop on April 11 in honor of Leszek Demkowicz’s 70th Birthday.

Presentations spanned topics related to Leszek’s life work

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Abel Prize Goes to Mathematician Who Studied Equations That Describe Nature

Media Coverage

March 22, 2023

Abel Prize Goes to Mathematician Who Studied Equations That Describe Nature

Read more at nytimes

Mathematician wins Abel Prize for 'smooth' physics

Media Coverage

March 22, 2023

Mathematician wins Abel Prize for 'smooth' physics

Read more at nature

Mathematics’ Highest Prize Awarded to Luis Caffarelli

News

March 22, 2023

Mathematics’ Highest Prize Awarded to Luis Caffarelli

Winner of the 2023 Abel Prize, considered the Nobel Prize equivalent in mathematics. 

He is the first Latin American mathematician to receive the award and the second in four years from UT Austin.

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SIAM Establishes the Renata Babuška Prize

News

Sept. 22, 2022

SIAM Establishes the Renata Babuška Prize

  • Named in memory of Dr. Ivo Babuška’s late wife, Renata, who passed away in 2020
  • The prize will be awarded in recognition of high-quality interdisciplinary work targeting modeling and numerical solutions of a specific engineering or scientific application

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Advancing Interdisciplinary Data Science Research

News

Aug. 18, 2022

Advancing Interdisciplinary Data Science Research

  • Award brings together scientists and engineers from different research communities to further the theoretical foundations of data science
  • Researchers aim to combat data sets that include historical biases and elements influenced by self-interested and possibly malicious parties

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Postdoc Takes Home the 2022 Householder Prize

News

July 21, 2022

Postdoc Takes Home the 2022 Householder Prize

  • Householder Prize recognizes best numerical linear algebra dissertation produced within the past three years 
  • Second award in 14 months for Heather Wilber's ‘Computing Numerically with Rational Functions’

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Oden Institute Recognizes 2021-2022 Graduates

News

June 16, 2022

Oden Institute Recognizes 2021-2022 Graduates

  • Event honored 16 students that graduated in May 2022, as well as eight other students who graduated throughout the 2021-2022 academic year
  • "Congratulations to all our PhD and Master's graduate students. Collectively, you represent an incredible set of accomplishments spanning computing, applied mathematics, statistics, machine learning, engineering, science, geoscience, and medicine."

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Distinguished Researcher Award 2022 Goes to Rachel Ward

News

June 13, 2022

Distinguished Researcher Award 2022 Goes to Rachel Ward

  • Award recognizes Oden Institute core faculty who have demonstrated a sustained record of distinguished research in computational engineering and sciences
  • Ward has made significant contributions to stochastic gradient descent, compressive sensing and randomized linear embeddings

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Inaugural Babuška Scholarship Recipient Announced

News

May 31, 2022

Inaugural Babuška Scholarship Recipient Announced

  • New scholarship funded by the Ivo & Renata Babuška Endowed Excellence Fund
  • Bronson Zhou is most drawn towards probability, algebra and the use of applied mathematics in neuroscience research

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Oden Institute Awards Ceremony 2022

Feature

May 18, 2022

Oden Institute Awards Ceremony 2022

  • Awards Ceremony finally held in-person after two years of hiatus from COVID  
  • Staff, students and faculty celebrated and recognized for their significant contributions throughout the year

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Tackling Science’s Grand Challenges

News

May 18, 2022

Tackling Science’s Grand Challenges

  • Five successful applicants gain the opportunity to tackle some of society's most pressing challenges 
  • Researchers will take on a broad range of challenges - from water efficiencies, materials science at the exascale,  the development of more energy-efficient algorithms, as well as advancing individualized care for breast cancer patients     

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Outstanding Dissertation Award Winner 2022 - Gopal Yalla

News

April 14, 2022

Outstanding Dissertation Award Winner 2022 - Gopal Yalla

  • Project outlining improvements in turbulence modeling of major signficance to those studying "the most important unsolved problem of classical physics" 
  • PhD graduate Gopal Yalla designs filters using underlying numerical operators explicitly

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A Lifelong Researcher - Robert van de Geijn

Feature

April 13, 2022

A Lifelong Researcher - Robert van de Geijn

  • Pivotal figure at the Oden institute may have retired from UT Austin but still as busy as ever
  • "For a computer scientist - a true computer scientist - it is all about exposing the system”

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Barbara Wohlmuth Elected to German National Academy of Sciences

News

April 12, 2022

Barbara Wohlmuth Elected to German National Academy of Sciences

  • Significant honor puts Wohlmuth in the company of scientific figures like Albert Einstein, Max Planck, Charles Darwin and Marie Curie
  • Her election to the German National Academy of Sciences - as a computational applied mathematician – establishes the field as a principal core area for scientific knowledge

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Gregory J. Rodin Retires

Feature

April 5, 2022

Gregory J. Rodin Retires

  • Newly retired Professor Emeritus of Aerospace Engineering and Core Faculty at the Oden Institute, Gregory Rodin is an expert in the field of computational micromechanics
  • He is also a big advocate for scientists trusting their own intuition

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Math for Poets - Profile Heather Wilber

Profile

Dec. 1, 2021

Math for Poets - Profile Heather Wilber

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Oden Institute Joins the Forty for 40 Party

News

Sept. 22, 2021

Oden Institute Joins the Forty for 40 Party

  • First time Oden Institute participates in 40 hours for the Forty Acres  
  • Annual UT Austin fundraising event now in its ninth year

Read more at utexas

Thomas Hughes Honored by Japan Association for Computational Mechanics

News

Aug. 30, 2021

Thomas Hughes Honored by Japan Association for Computational Mechanics

  • Peter O'Donnell, Jr. Chair in Computational and Applied Mathematics inducted with Tayfun Tezduyar from Rice University
  • Tezduyar and Hughes first non-Japanese so honored by JACM

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Preschoolers, PhDs and The Pandemic - Profile Ian & Amelia Henrikksen

Profile

April 27, 2021

Preschoolers, PhDs and The Pandemic - Profile Ian & Amelia Henrikksen

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Tan Bui-Thanh Elected SIAM Secretary

News

Feb. 4, 2021

Tan Bui-Thanh Elected SIAM Secretary

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