# Adaptive High-Order Methods for Fluid Flows Authors: Lott Adaptive High-Order Methods for Fluid Flows
Authors: Lott & Deane

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Applied Mathematics and Scientific Computation
The goal of this research is to construct an Adaptive Spectral Element Method to solve non-linear PDEs.
We aim to create an efficient method for solving large scale PDE's on complex domains. We use adaptivity to maintain a high working accuracy for long time periods without wasting computational resources. Spectral Element
Methods are computationally efficient for higher dimensional problems due to their tensor product formulation of the system operators.

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The Spectral Element Method approximates the solution to the PDE by
partitioning the domain into elements, and on each element, solving an
integral equation. In order to solve the equation, one constructs a
weighted sum of polynomial basis functions. In particular, SEM uses the
Gauss-Lobatto-Legendre quadrature rules in order to obtain an exact
solution for integrands of order 2P+1 or less, when using Legendre
polynomials of degree P. This gives SEM an exponential convergence
property that methods such as Finite Element, or Finite Difference
methods dont have. As we increase the polynomial degree in the SEM,
we exponentially approach the solution to the integral equation. In low
order methods, where only the number of elements is increased, one
obtains only algebraic convergence to the solution.

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Solution to 1D Viscous Burgers Equation using P-Adaptivity
We have implemented error estimators that
determine when to increase or decrease the
polynomial degree on each element in order to
obtain an accurate solution without wasting
computational resources in well behaved areas
of the solution.
To the right, we show the computational work
required to integrate a linear advection equation
Slope of solution at x=0, within 5 periods while maintaining a phase shift error