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Copy file name to clipboardExpand all lines: docs/src/background.md
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# Mathematical Background
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Local Fourier Analysis (LFA) was first used by Brandt [1] to analyze the convergence of multi-level adaptive techniques for solving PDEs discretized with finite differences, but the technique has been adapted for multi-level and multi-grid techniques using finite element discretizations.
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Local Fourier Analysis (LFA) was first used in [brandt1977](@cite) to analyze the convergence of multi-level adaptive techniques for solving PDEs discretized with finite differences, but the technique has been adapted for multi-level and multi-grid techniques using finite element discretizations.
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While this library focuses on the finite element discretizations, finite difference discretizations of PDEs can often be recovered from finite element formulations by using linear finite elements on a structured grid.
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This fact makes LFAToolkit.jl an extremely flexible tool for LFA.
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## High Order Finite Elements
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Consider the specific case of a Topeliz operator representing a scalar PDE in 1D with the weak formulation given by Brown in [2],
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Consider the specific case of a Topeliz operator representing a scalar PDE in 1D with the weak formulation given by [brown2010](@cite),
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```math
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\int_{\Omega} v \cdot f_0 \left( u, \nabla u \right) + \nabla v : f_1 \left( u, \nabla u \right) = \int_{\Omega} f v, \forall v \in V
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A u = b.
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```
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Using the algebraic representation of PDE operators discussed in [2], the PDE operator ``A`` is of the form
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Using the algebraic representation of PDE operators discussed in [brown2010](@cite), the PDE operator ``A`` is of the form
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```math
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A = P^T A_e P
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where ``\nu`` is the number of smoothing passes.
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More sophisticated smoothers can be used, such as the Chebyshev semi-iterative method.
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For discussion of the error propegation of the Chebyshev semi-iteative method, see Gutknecht and Röllin [3].
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For discussion of the error propegation of the Chebyshev semi-iteative method, see [gutknecht2002](@cite).
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User defined smoothers are supported, where the user provides ``M^{-1}`` or a function computing ``M^{-1}`` based upon ``A``, and ``\tilde{M}^{-1}_h`` and ``\tilde{S}_h`` are automatically generated and used inside the multigrid symbol matrix.
Copy file name to clipboardExpand all lines: docs/src/examples/advection.md
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The advection operator is a classical test case to see dispersion spectrum inside LFAToolkit.
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Here we show the advection operator on a non-polynomial basis derived from the Hale-Trefethen strip transformation applied to a H1 Lagrange basis.
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For understanding about nonpolynomial bases, see paper Hale and Trefethen (2008) New quadrature formulas from conformal maps. https://doi.org/10.1137/07068607X
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For further discussion on non-polynomial bases, see [hale2008](@cite).
Copy file name to clipboardExpand all lines: docs/src/examples/advection_supg.md
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In this weak formulation, boundary terms have been omitted, as they are not present on the infinite grid for Local Fourier Analysis.
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The SUPG stabilization is controlled by the parameter ``τ``, where ``τ = 0`` gives the classical Galerkin formulation and ``τ = \dfrac{h}{2}`` gives a nodally exact solution to the steady advection equation with source when using linear elements (this can be extended to advection-diffusion with a further scaling that depends on the cell Péclet number).
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For discussion on SUPG, see [hughes1979](@cite), [brooks1982](@cite), and [whiting2003](@cite).
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### LFAToolkit code
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The symbol of the continuous advection operator ``u_t + c u_x = 0`` applied to the Fourier mode ``e^{i\theta x}`` is ``i\theta``.
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To understand dispersion within the resolved frequencies, we instead plot the phase speed ``\lambda/\theta``, which should be very close to ``c`` through the resolved frequencies.
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Here we show the SUPG advection operator on ``H^1`` Lagrange basis.
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For understanding about SUPG in this work, see papers by Hughes TJR, Brooks AN (1979, 1982) and C.H. Whiting
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A multi-dimensional upwind scheme with no crosswind diffusion. In: Hughes TJR, editor. Finite element methods for convection dominated flows, AMD-vol. 34. New York: ASME, (1979), pp. 19-35.
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Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Meth Appl Mech Eng, 32 (1982), pp. 199-259.
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Hierarchical basis for stabilized finite element methods for compressible flows. Comput. Methods Appl. Mech. Engrg. 192, (2003), pp. 5167-5185.
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