Discontinuous Galerkin Methods

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Discontinuous Galerkin Methods

• Li, Yang
• FerienAkademie 2008

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Contents
Methods of solving PDEs
Introduction of DG Methods
Working with 1-Dimension
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Methods of solving PDEs
Finite Difference Method
PDEs
Finite Volume Method
Finite Element Method
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Methods of solving PDEs

• E.g. 1D scalar conservation law
• with initial conditions and boundary conditions
  on the boundary
• unknown solution
• flux
• prescribed force

How to get the approximate solution ? Is it
satisfied the equation?
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Methods of solving PDEs

• Finite Difference Method
• a grid
• local grid size
• assume
• Residual

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Methods of solving PDEs
Element-based discretization to ensure geometry
flexibility

• Finite Difference
• Method

? Simple to implement
? Ill-suited to deal with complex geometries
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Methods of solving PDEs

• Finite Volume Method
• element staggered grid ,
• solution is approximated on the element by a
  constant

The actual numerical scheme will depend upon
problem geometry and mesh construction. Difficult
when high-order reconstruction.
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Methods of solving PDEs

• Finite Element Method
• assume the local solution
• element locally defined basis
  function
• global representation of
• where is the basis function.
• define a space of test functions, , and
  require the residual is orthogonal to all test
  functions

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Methods of solving PDEs

• Finite Element Method
• Classical choice the spaces spanned by the
  basis functions and test functions are the same.
• Since the residual has to vanish for all

? Easy to extend to high-order approximation by
adding additional degrees of freedom to the
element.
? The semi-discrete scheme becomes implicit and M
must be inverted
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Introduction of DG Methods

• The Discontinuous Galerkin method is somewhere
  between a finite element and a finite volume
  method and has many good features of both,
  utilizing a space of basis and test functions
  that mimics the finite element method but
  satisfying the equation in a sense closer to the
  finite volume method.
• It provides a practical framework for the
  development of high-order accurate methods using
  unstructured grids. The method is well suited for
  large-scale time-dependent computations in which
  high accuracy is required.
• An important distinction between the DG method
  and the usual finite-element method is that in
  the DG method the resulting equations are local
  to the generating element. The solution within
  each element is not reconstructed by looking to
  neighboring elements. Its compact formulation can
  be applied near boundaries without special
  treatment, which greatly increases the robustness
  and accuracy of any boundary condition
  implementation. 

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Introduction of DG Methods

• From FEM and FVM to DG-FEM
• maintain the definition of elements as in the
  FEM
• but new definition of vector of unknowns
• Assume the local solution in each element is
  (likewise for the flux)
• Define The space of basis functions
• The local residual is

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Introduction of DG Methods

• Require that the residual is orthogonal to all
  test functions
• Similar to FVM, use Gauss theorem
• introduce the numerical flux, , as the
  unique value to be used at the interface and
  obtained by coming information from both
  elements.
• applying Gauss theorem again

Weak Form
Strong Form
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Introduction of DG Methods

• More general form
• Consider the nonlinear, scalar, conservation
  law
• subject to appropriate initial conditions
• The boundary conditions are provided when the
  boundary is an inflow boundary
• when
• when
• We still assume that the global solution can be
  well approximated by a space of piecewise
  polynomial functions, defined on the union of
  , and require the residual to be orthogonal to
  space of the test functions,

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Introduction of DG Methods

• recover the locally defined weak formulation
• and the strong form
• Assume that all local test functions can be
  represented by using a local polynomial basis,
  , as
• and leads to equations as

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Working with 1-Dimension

• E.g.
• Choose the basis functions Jacobi polynomials
• Integral Gaussian quadrature
• Time 4th order explicit RK method
• Simple algorithm steps
• Generate simple mesh
• Construct the matrices
• Solve the equation system

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Working with 1-Dimension
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Working with 1-Dimension
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Reference

• Jan S Hesthaven, Tim Warburton Nodal
  Discontinuous Galerkin Methods Algorithms,
  Analysis, and Applications, Springer
• Cockburn B, Shu CW   TVB Runge-Kutta local
  projection discontinuous Galerkin finite element
  method for conservation laws II general
  framework,   MATHEMATICS OF COMPUTATION, v52
  (1989), pp.411-435.
• http//lsec.cc.ac.cn/lcfd/DGM_mem.html
• http//www.wikipedia.org/
• http//www.nudg.org/

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Thank You !