- · LFAToolkit.jl

Finite Element Operator Gallery

The following PDEs are available in the operator gallery.

Scalar Mass, Diffusion, Advection, and SUPG Advection Operators

LFAToolkit.GalleryOperator — Function

GalleryOperator(
    name,
    numbernodes1d,
    numberquadraturepoints1d,
    mesh;
    collocatedquadrature = false,
    mapping = nothing,
    parameters = nothing,
)

Finite element operator from a gallery of options

Arguments:

name::String: string containing name of operator
numbernodes1d::Int: polynomial order of TensorH1LagrangeBasis
numberquadraturepoints1d::Int: number of quadrature points in one dimension for basis
mesh::Mesh: mesh for operator

Keyword Arguments:

collocatedquadrature::Bool = false: Gauss-Legendre or Gauss-Legendre-Lobatto quadrature points
mappingUnion{Tuple{Function,Function},Nothing} = nothing: quadrature point mapping - sausage, Kosloff-Talezer, Hale-Trefethen strip, or no transformation
parametersUnion{NamedTuple,Nothing} = nothing: named tuple of model parameters

Returns:

finite element operator object

Mass matrix example:

# setup
mesh = Mesh2D(1.0, 1.0);
mass = GalleryOperator("mass", 4, 4, mesh);

# verify
println(mass)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    interpolation
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    interpolation

Diffusion operator example:

# setup
mesh = Mesh2D(1.0, 1.0);
diffusion = GalleryOperator("diffusion", 4, 4, mesh);

# verify
println(diffusion)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    gradient
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    gradient

Advection operator example:

# setup
mesh = Mesh2D(1.0, 1.0);
advection = GalleryOperator("advection", 4, 4, mesh);

# verify
println(advection)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    interpolation
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    gradient

SUPG mass matrix example:

# setup
mesh = Mesh2D(1.0, 1.0);
parameters = (wind = [1.0, 1.0], τ = 1.0);
supgmass = GalleryOperator("supg mass", 4, 4, mesh, parameters = parameters);

# verify
println(supgmass)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    interpolation
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation modes:
    interpolation
    gradient

SUPG advection operator example:

# setup
mesh = Mesh2D(1.0, 1.0);
parameters = nothing;
supgadvection = GalleryOperator("supg advection", 4, 4, mesh, parameters = parameters);

# verify
println(supgadvection)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation modes:
    interpolation
    gradient
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    gradient

source

Vector Mass and Diffusion Operators

LFAToolkit.GalleryVectorOperator — Function

GalleryVectorOperator(name, numbernodes1d, numberquadraturepoints1d, numberelements1d, mesh)

Finite element operator from a gallery of options

Arguments:

name::String: string containing name of operator
numbernodes1d::Int: polynomial order of TensorH1LagrangeBasis
numberquadraturepoints1d::Int: number of quadrature points in one dimension for basis
numbercomponents::Int: number of components
mesh::Mesh: mesh for operator

Returns:

finite element operator object

Mass matrix example:

# setup
mesh = Mesh2D(1.0, 1.0);
mass = GalleryVectorOperator("mass", 4, 4, 3, mesh);

# verify
println(mass)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 3
    dimension: 2
  evaluation mode:
    interpolation
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 3
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 3
    dimension: 2
  evaluation mode:
    interpolation

Diffusion operator example:

# setup
mesh = Mesh2D(1.0, 1.0);
diffusion = GalleryVectorOperator("diffusion", 4, 4, 3, mesh);

# verify
println(diffusion)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 3
    dimension: 2
  evaluation mode:
    gradient
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 3
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 3
    dimension: 2
  evaluation mode:
    gradient

source

Scalar Macro-Element Mass and Diffusion Operators

LFAToolkit.GalleryMacroElementOperator — Function

GalleryMacroElementOperator(
    name,
    numbernodes1d,
    numberquadraturepoints1d,
    numberelements1d,
    mesh;
    mapping = nothing,
    parameters = nothing,
)

Finite element operator from a gallery of options

Arguments:

name::String: string containing name of operator
numbernodes1d::Int: polynomial order of TensorH1LagrangeBasis
numberquadraturepoints1d::Int: number of quadrature points in one dimension for basis
numberelements1d::Int: number of elements in macro-element
mesh::Mesh: mesh for operator

Keyword Arguments:

mapping::Union{Tuple{Function,Function},Nothing} = nothing: quadrature point mapping - sausage, Kosloff-Talezer, Hale-Trefethen strip, or no transformation
parameters::Union{NamedTuple,Nothing} = nothing: named tuple of model parameters

Returns:

finite element operator object

Mass matrix example:

# setup
mesh = Mesh2D(1.0, 1.0);
mass = GalleryMacroElementOperator("mass", 4, 4, 2, mesh);

# verify
println(mass)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  macro-element tensor product basis:
    numbernodes1d: 7
    numberquadraturepoints1d: 8
    numbercomponents: 1
    numberelements1d: 2
    dimension: 2
  evaluation mode:
    interpolation
operator field:
  macro-element tensor product basis:
    numbernodes1d: 7
    numberquadraturepoints1d: 8
    numbercomponents: 1
    numberelements1d: 2
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  macro-element tensor product basis:
    numbernodes1d: 7
    numberquadraturepoints1d: 8
    numbercomponents: 1
    numberelements1d: 2
    dimension: 2
  evaluation mode:
    interpolation

Diffusion operator example:

# setup
mesh = Mesh2D(1.0, 1.0);
diffusion = GalleryMacroElementOperator("diffusion", 4, 4, 2, mesh);

# verify
println(diffusion)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  macro-element tensor product basis:
    numbernodes1d: 7
    numberquadraturepoints1d: 8
    numbercomponents: 1
    numberelements1d: 2
    dimension: 2
  evaluation mode:
    gradient
operator field:
  macro-element tensor product basis:
    numbernodes1d: 7
    numberquadraturepoints1d: 8
    numbercomponents: 1
    numberelements1d: 2
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  macro-element tensor product basis:
    numbernodes1d: 7
    numberquadraturepoints1d: 8
    numbercomponents: 1
    numberelements1d: 2
    dimension: 2
  evaluation mode:
    gradient

Advection operator example:

# setup
mesh = Mesh2D(1.0, 1.0);
advection = GalleryVectorOperator("advection", 4, 4, 1, mesh);

# verify
println(advection)

# output

finite element operator:
2d mesh:
    dx: 1.0
    dy: 1.0

2 inputs:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    interpolation
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    quadratureweights

1 output:
operator field:
  tensor product basis:
    numbernodes1d: 4
    numberquadraturepoints1d: 4
    numbercomponents: 1
    dimension: 2
  evaluation mode:
    gradient

source