Hodge Laplacian

Introduction

In this example, ww solve the Hodge-Laplacian for $0$-forms with Dirichlet boundary conditions. The $0$-form Hodge-Laplacian on a domain $\mathrm{\Omega }$ with boundary $\partial \mathrm{\Omega }$ is

$$\begin{array}{rlrl}& {\mathrm{d}}^{\ast }\mathrm{d}{\varphi }^0=-f^0& & \text{on}\mathrm{\Omega },\\ & tr({\varphi }^0)=0& & \text{on}\partial \mathrm{\Omega }.\end{array}$$

The weak formulation is then as follows.

$$\begin{array}{c}\text{Given}{f}^0\in L^2{\mathrm{\Lambda }}^0(\mathrm{\Omega }),\text{find}{\varphi }^0\in {\mathrm{\Lambda }}_h^0(\mathrm{\Omega })\text{such that}\\ {\int }_{\mathrm{\Omega }}\mathrm{d}{\varphi }^0\wedge \star \mathrm{d}{\phi }^0={\int }_{\mathrm{\Omega }}{f}^0\wedge \star {\phi }^0\mathrm{\forall }{\phi }^0\in {\mathrm{\Lambda }}_{h,0}^0(\mathrm{\Omega }).\end{array}$$

Implementation

We start by defining the geometry and the function space. The geometry is a simple unit square with $4\times 4$ elements. The function space is a B-spline space with degree $p=(3,3)$ and knot vector $k=(2,2)$.

using Mantis

starting_point = (0.0, 0.0)
box_size = (1.0, 1.0)
num_elements = (4, 4)
geometry = Geometry.create_cartesian_box(starting_point, box_size, num_elements)

p = (3, 3)
k = (2, 2)
B = FunctionSpaces.create_bspline_space(starting_point, box_size, num_elements, p, k)

Mantis.FunctionSpaces.TensorProductSpace{2, 1, 1, 2, Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, Vector{Vector{Vector{Int64}}}}((Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]]), Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]])), Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), CartesianIndices((7, 7)), LinearIndices((Base.OneTo(7), Base.OneTo(7))), [[[1], [2, 3, 4, 5, 6], [7], [8, 15, 22, 29, 36], [9, 10, 11, 12, 13, 16, 17, 18, 19, 20  …  30, 31, 32, 33, 34, 37, 38, 39, 40, 41], [14, 21, 28, 35, 42], [43], [44, 45, 46, 47, 48], [49]]])

Mantis works with forms, so we need to define the form space. In this case, we are working with $0$-forms, so we define the form space as follows.

Λ⁰ = Forms.FormSpace(0, B, "label")

Mantis.Forms.FormSpace{2, 0, Mantis.FunctionSpaces.TensorProductSpace{2, 1, 1, 2, Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, Vector{Vector{Vector{Int64}}}}, String}(Mantis.FunctionSpaces.TensorProductSpace{2, 1, 1, 2, Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, Vector{Vector{Vector{Int64}}}}((Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]]), Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]])), Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), CartesianIndices((7, 7)), LinearIndices((Base.OneTo(7), Base.OneTo(7))), [[[1], [2, 3, 4, 5, 6], [7], [8, 15, 22, 29, 36], [9, 10, 11, 12, 13, 16, 17, 18, 19, 20  …  30, 31, 32, 33, 34, 37, 38, 39, 40, 41], [14, 21, 28, 35, 42], [43], [44, 45, 46, 47, 48], [49]]]), "label")

We define the weak form inputs. The weak form inputs contain the trial and test spaces, the forcing function, and the quadrature rule. We define the forcing function as a function of the coordinates. In this case, we define the forcing function as $f^0=8{\pi }^2\sin (2\pi x)\sin (2\pi y)$.

function forcing_function(x::Matrix{Float64})
    return [@. 8.0 * pi^2 * sin(2.0 * pi * x[:, 1]) * sin(2.0 * pi * x[:, 2])]
end
f⁰ = Forms.AnalyticalFormField(0, forcing_function, geometry, "f⁰")

Mantis.Forms.AnalyticalFormField{2, 0, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, typeof(Main.forcing_function), String}(Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), Main.forcing_function, "f⁰")

The quadrature rule is defined as a tensor product rule of the degree of the B-spline space plus one. In this case, we define the quadrature rule as a Gauss-Legendre rule.

canonical_qrule = Quadrature.tensor_product_rule(p .+ 1, Quadrature.gauss_legendre)
dΩ = Quadrature.StandardQuadrature(canonical_qrule, Geometry.get_num_elements(geometry))

Mantis.Quadrature.StandardQuadrature{2, Mantis.Quadrature.CanonicalQuadratureRule{2, Mantis.Points.CartesianPoints{2, Vector{Float64}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}}(Mantis.Quadrature.CanonicalQuadratureRule{2, Mantis.Points.CartesianPoints{2, Vector{Float64}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(Mantis.Points.CartesianPoints{2, Vector{Float64}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}(([0.06943184420297371, 0.33000947820757187, 0.6699905217924281, 0.9305681557970262], [0.06943184420297371, 0.33000947820757187, 0.6699905217924281, 0.9305681557970262]), CartesianIndices((4, 4)), LinearIndices((Base.OneTo(4), Base.OneTo(4)))), [0.0302507483214005, 0.056712962962962965, 0.056712962962962965, 0.0302507483214005, 0.056712962962962965, 0.10632332575267359, 0.10632332575267359, 0.056712962962962965, 0.056712962962962965, 0.10632332575267359, 0.10632332575267359, 0.056712962962962965, 0.0302507483214005, 0.056712962962962965, 0.056712962962962965, 0.0302507483214005], "Tensor-product of (Gauss-Legendre, Gauss-Legendre) rules"), 16)

We define the weak form inputs as a WeakFormInputs object. The weak form inputs contain the trial and test spaces, and the forcing function. The trial and test spaces are the same in this case, which is the default.

wfi = Assemblers.WeakFormInputs(Λ⁰, f⁰)

Mantis.Assemblers.WeakFormInputs{2, Tuple{Mantis.Forms.FormSpace{2, 0, Mantis.FunctionSpaces.TensorProductSpace{2, 1, 1, 2, Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, Vector{Vector{Vector{Int64}}}}, String}}, Tuple{Mantis.Forms.FormSpace{2, 0, Mantis.FunctionSpaces.TensorProductSpace{2, 1, 1, 2, Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, Vector{Vector{Vector{Int64}}}}, String}}, Tuple{Mantis.Forms.AnalyticalFormField{2, 0, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, typeof(Main.forcing_function), String}}}((Mantis.Forms.FormSpace{2, 0, Mantis.FunctionSpaces.TensorProductSpace{2, 1, 1, 2, Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, Vector{Vector{Vector{Int64}}}}, String}(Mantis.FunctionSpaces.TensorProductSpace{2, 1, 1, 2, Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}, CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}, Vector{Vector{Vector{Int64}}}}((Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]]), Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]])), Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), CartesianIndices((7, 7)), LinearIndices((Base.OneTo(7), Base.OneTo(7))), [[[1], [2, 3, 4, 5, 6], [7], [8, 15, 22, 29, 36], [9, 10, 11, 12, 13, 16, 17, 18, 19, 20  …  30, 31, 32, 33, 34, 37, 38, 39, 40, 41], [14, 21, 28, 35, 42], [43], [44, 45, 46, 47, 48], [49]]]), "label"),), (Mantis.Forms.FormSpace{2, 0, Mantis.FunctionSpaces.TensorProductSpace{2, 1, 1, 2, Tuple{Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}, Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, 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We define the weak form for the Hodge-Laplacian. The weak form is defined as a function that takes the weak form inputs and the quadrature rule as arguments.

function zero_form_hodge_laplacian(
    inputs::Assemblers.AbstractInputs, dΩ::Quadrature.AbstractGlobalQuadratureRule
)
    v⁰ = Assemblers.get_test_form(inputs)
    u⁰ = Assemblers.get_trial_form(inputs)
    f⁰ = Assemblers.get_forcing(inputs)

    A = ∫(d(v⁰) ∧ ★(d(u⁰)), dΩ)
    lhs_expression = ((A,),)

    b = ∫(v⁰ ∧ ★(f⁰), dΩ)
    rhs_expression = ((b,),)

    return lhs_expression, rhs_expression
end

zero_form_hodge_laplacian (generic function with 1 method)

We can now assemble the linear system and solve it to obtain the solution. We define the Dirichlet boundary conditions using the appropriate helper function.

bc = Forms.set_dirichlet_boundary_conditions(Λ⁰, 0.0)

Dict{Int64, Float64} with 24 entries:
  5  => 0.0
  35 => 0.0
  28 => 0.0
  8  => 0.0
  1  => 0.0
  22 => 0.0
  47 => 0.0
  6  => 0.0
  43 => 0.0
  45 => 0.0
  36 => 0.0
  44 => 0.0
  49 => 0.0
  14 => 0.0
  3  => 0.0
  29 => 0.0
  7  => 0.0
  46 => 0.0
  48 => 0.0
  ⋮  => ⋮

We assemble the linear system and solve it to obtain the solution.

lhs_expressions, rhs_expressions = zero_form_hodge_laplacian(wfi, dΩ)
weak_form = Assemblers.WeakForm(lhs_expressions, rhs_expressions, wfi)
A, b = Assemblers.assemble(weak_form, bc)
sol = vec(A \ b)
ϕ⁰ = Forms.build_form_field(Λ⁰, sol)

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UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]]), Mantis.FunctionSpaces.BSplineSpace{Mantis.FunctionSpaces.Bernstein, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}, Vector{Vector{Vector{Int64}}}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), Mantis.FunctionSpaces.KnotVector{Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}, Vector{Int64}}(Mantis.Geometry.CartesianGeometry{1, 1, 1, Tuple{Tuple{LinRange{Float64, Int64}}}, Tuple{CartesianIndices{1, Tuple{Base.OneTo{Int64}}}}, Tuple{LinearIndices{1, Tuple{Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5),),), (CartesianIndices((4,)),), (LinearIndices((Base.OneTo(4),)),)), 3, [4, 1, 1, 1, 4]), Mantis.FunctionSpaces.Bernstein(3), Mantis.FunctionSpaces.ExtractionOperator{1, Matrix{Float64}, UnitRange{Int64}, UnitRange{Int64}}([([1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.5 0.5 0.0; 0.0 0.25 0.5833333333333334 0.16666666666666666],), ([0.25 0.5833333333333334 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.6666666666666667 0.16666666666666666],), ([0.16666666666666669 0.6666666666666667 0.16666666666666666 0.0; 0.0 0.6666666666666667 0.3333333333333333 0.0; 0.0 0.33333333333333337 0.6666666666666666 0.0; 0.0 0.16666666666666669 0.5833333333333333 0.25],), ([0.16666666666666669 0.5833333333333333 0.25 0.0; 0.0 0.5 0.5 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0],)], Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}[Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(1:4, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(2:5, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(3:6, (1:4,)), Mantis.FunctionSpaces.Indices{1, UnitRange{Int64}, UnitRange{Int64}}(4:7, (1:4,))], 4, 7), [[[1], [2, 3, 4, 5, 6], [7]]])), Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), Mantis.Geometry.CartesianGeometry{2, 2, 1, Tuple{Tuple{LinRange{Float64, Int64}, LinRange{Float64, Int64}}}, Tuple{CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}, Tuple{LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}}}(((LinRange{Float64}(0.0, 1.0, 5), LinRange{Float64}(0.0, 1.0, 5)),), (CartesianIndices((4, 4)),), (LinearIndices((Base.OneTo(4), Base.OneTo(4))),)), CartesianIndices((7, 7)), LinearIndices((Base.OneTo(7), Base.OneTo(7))), [[[1], [2, 3, 4, 5, 6], [7], [8, 15, 22, 29, 36], [9, 10, 11, 12, 13, 16, 17, 18, 19, 20  …  30, 31, 32, 33, 34, 37, 38, 39, 40, 41], [14, 21, 28, 35, 42], [43], [44, 45, 46, 47, 48], [49]]]), "label"), [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.27665525556726794, 0.7935323680513704  …  0.7935323680513695, 0.27665525556726844, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], "label")

We can now plot the solution using the plot function. This will write the output to a VTK file that can be visualized using a VTK viewer, such as Paraview.

data_folder = joinpath(dirname(dirname(pathof(Mantis))), "examples", "data")
output_data_folder = joinpath(data_folder, "output", "HodgeLaplacian")
output_filename = "HodgeLaplacian-0form-Dirichlet-2D.vtu"
output_file = joinpath(output_data_folder, output_filename)
Plot.plot(
    ϕ⁰;
    vtk_filename=output_file,
    n_subcells=1,
    degree=maximum(p),
    ascii=false,
    compress=false,
)

1-element Vector{String}:
 "/home/runner/work/Mantis.jl/Man" ⋯ 60 bytes ⋯ "orm-Dirichlet-2D_wireframe.vtu"

Extensions

3D

We can extend this example to 3D by defining the geometry as a unit cube with $4\times 4\times 4$ elements. We define the function space as a B-spline space with degree $p=(3,3,3)$ and knot vector $k=(2,2,2)$. We define the forcing function as $f^0=12{\pi }^2\sin (2\pi x)\sin (2\pi y)\sin (2\pi z)$. Note that the new variables are renamed to avoid overwriting the previous variables and to avoid type instabilities.

starting_point_3D = (0.0, 0.0, 0.0)
box_size_3D = (1.0, 1.0, 1.0)
num_elements_3D = (4, 4, 4)
geometry_3D = Geometry.create_cartesian_box(starting_point_3D, box_size_3D, num_elements_3D)

p_3D = (3, 3, 3)
k_3D = (2, 2, 2)
B_3D = FunctionSpaces.create_bspline_space(
    starting_point_3D, box_size_3D, num_elements_3D, p_3D, k_3D
)

function forcing_function_3D(x::Matrix{Float64})
    return [
        @. 12.0 *
            pi^2 *
            sin(2.0 * pi * x[:, 1]) *
            sin(2.0 * pi * x[:, 2]) *
            sin(2.0 * pi * x[:, 3])
    ]
end

forcing_function_3D (generic function with 1 method)

We don't need to change any of the previous code, so we can reuse it directly:

Λ⁰_3D = Forms.FormSpace(0, FunctionSpaces.DirectSumSpace((B_3D,)), "label")

f⁰_3D = Forms.AnalyticalFormField(0, forcing_function_3D, geometry_3D, "f⁰")

canonical_qrule_3D = Quadrature.tensor_product_rule(p_3D .+ 1, Quadrature.gauss_legendre)
dΩ_3D = Quadrature.StandardQuadrature(
    canonical_qrule_3D, Geometry.get_num_elements(geometry_3D)
)

wfi_3D = Assemblers.WeakFormInputs(Λ⁰_3D, f⁰_3D)

bc_3D = Forms.set_dirichlet_boundary_conditions(Λ⁰_3D, 0.0)

lhs_expressions_3D, rhs_expressions_3D = zero_form_hodge_laplacian(wfi_3D, dΩ_3D)
weak_form_3D = Assemblers.WeakForm(lhs_expressions_3D, rhs_expressions_3D, wfi_3D)
A_3D, b_3D = Assemblers.assemble(weak_form_3D, bc_3D)
sol_3D = vec(A_3D \ b_3D)
ϕ⁰_3D = Forms.build_form_field(Λ⁰_3D, sol_3D)

output_filename_3D = "HodgeLaplacian-0form-Dirichlet-3D.vtu"
output_file_3D = joinpath(output_data_folder, output_filename_3D)
Plot.plot(
    ϕ⁰_3D;
    vtk_filename=output_file_3D,
    n_subcells=1,
    degree=maximum(p_3D),
    ascii=false,
    compress=false,
)

1-element Vector{String}:
 "/home/runner/work/Mantis.jl/Man" ⋯ 60 bytes ⋯ "orm-Dirichlet-3D_wireframe.vtu"

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