--- a/src/SbpOperators/volumeops/laplace/laplace.jl	Mon May 23 07:20:27 2022 +0200
+++ b/src/SbpOperators/volumeops/laplace/laplace.jl	Tue Feb 10 22:41:19 2026 +0100
@@ -1,9 +1,8 @@
 """
     Laplace{T, Dim, TM} <: LazyTensor{T, Dim, Dim}
 
-Implements the Laplace operator, approximating ∑d²/xᵢ² , i = 1,...,`Dim` as a
-`LazyTensor`. Additionally `Laplace` stores the `StencilSet`
-used to construct the `LazyTensor `.
+The Laplace operator, approximating ∑d²/xᵢ² , i = 1,...,`Dim` as a
+`LazyTensor`.
 """
 struct Laplace{T, Dim, TM<:LazyTensor{T, Dim, Dim}} <: LazyTensor{T, Dim, Dim}
     D::TM       # Difference operator
@@ -11,17 +10,15 @@
 end
 
 """
-    Laplace(grid::Equidistant, stencil_set)
+    Laplace(g::Grid, stencil_set::StencilSet)
 
-Creates the `Laplace` operator `Δ` on `grid` given a `stencil_set`. 
+Creates the `Laplace` operator `Δ` on `g` given `stencil_set`. 
 
 See also [`laplace`](@ref).
 """
-function Laplace(grid::EquidistantGrid, stencil_set::StencilSet)
-    inner_stencil = parse_stencil(stencil_set["D2"]["inner_stencil"])
-    closure_stencils = parse_stencil.(stencil_set["D2"]["closure_stencils"])
-    Δ = laplace(grid, inner_stencil,closure_stencils)
-    return Laplace(Δ,stencil_set)
+function Laplace(g::Grid, stencil_set::StencilSet)
+    Δ = laplace(g, stencil_set)
+    return Laplace(Δ, stencil_set)
 end
 
 LazyTensors.range_size(L::Laplace) = LazyTensors.range_size(L.D)
@@ -32,24 +29,99 @@
 # Base.show(io::IO, L::Laplace) = ...
 
 """
-    laplace(grid::EquidistantGrid, inner_stencil, closure_stencils)
-
-Creates the Laplace operator operator `Δ` as a `LazyTensor`
+    laplace(g::Grid, stencil_set)
 
-`Δ` approximates the Laplace operator ∑d²/xᵢ² , i = 1,...,`Dim` on `grid`, using
-the stencil `inner_stencil` in the interior and a set of stencils `closure_stencils`
-for the points in the closure regions.
+Creates the Laplace operator operator `Δ` as a `LazyTensor` on `g`.
 
-On a one-dimensional `grid`, `Δ` is equivalent to `second_derivative`. On a
-multi-dimensional `grid`, `Δ` is the sum of multi-dimensional `second_derivative`s
-where the sum is carried out lazily.
+`Δ` approximates the Laplace operator ∑d²/xᵢ² , i = 1,...,`Dim` on `g`. The
+approximation depends on the type of grid and the stencil set.
 
 See also: [`second_derivative`](@ref).
 """
-function laplace(grid::EquidistantGrid, inner_stencil, closure_stencils)
-    Δ = second_derivative(grid, inner_stencil, closure_stencils, 1)
-    for d = 2:dimension(grid)
-        Δ += second_derivative(grid, inner_stencil, closure_stencils, d)
+function laplace end
+function laplace(g::TensorGrid, stencil_set)
+    # return mapreduce(+, enumerate(g.grids)) do (i, gᵢ)
+    #     Δᵢ = laplace(gᵢ, stencil_set)
+    #     LazyTensors.inflate(Δᵢ, size(g), i)
+    # end
+
+    Δ = LazyTensors.inflate(laplace(g.grids[1], stencil_set), size(g), 1)
+    for d = 2:ndims(g)
+        Δ += LazyTensors.inflate(laplace(g.grids[d], stencil_set), size(g), d)
     end
     return Δ
 end
+laplace(g::EquidistantGrid, stencil_set) = second_derivative(g, stencil_set)
+
+"""
+    sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning, R_tuning)
+
+The operators required to construct the SAT for imposing a Dirichlet
+condition. `H_tuning` and `R_tuning` are used to specify the strength of the
+penalty.
+
+See also: [`sat`](@ref), [`DirichletCondition`](@ref), [`positivity_decomposition`](@ref).
+"""
+function sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning = 1., R_tuning = 1.)
+    id = boundary(bc)
+    set  = Δ.stencil_set
+    H⁻¹ = inverse_inner_product(g,set)
+    Hᵧ = inner_product(boundary_grid(g, id), set)
+    e = boundary_restriction(g, set, id)
+    d = normal_derivative(g, set, id)
+    B = positivity_decomposition(Δ, g, boundary(bc); H_tuning, R_tuning)
+    penalty_tensor = H⁻¹∘(d' - B*e')∘Hᵧ
+    return penalty_tensor, e
+end
+
+"""
+    sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
+
+The operators required to construct the SAT for imposing a Neumann condition.
+
+See also: [`sat`](@ref), [`NeumannCondition`](@ref).
+"""
+function sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
+    id = boundary(bc)
+    set  = Δ.stencil_set
+    H⁻¹ = inverse_inner_product(g,set)
+    Hᵧ = inner_product(boundary_grid(g, id), set)
+    e = boundary_restriction(g, set, id)
+    d = normal_derivative(g, set, id)
+
+    penalty_tensor = -H⁻¹∘e'∘Hᵧ
+    return penalty_tensor, d
+end
+
+"""
+    positivity_decomposition(Δ::Laplace, g::Grid, b::BoundaryIdentifier; H_tuning, R_tuning)
+
+Constructs the scalar `B` such that `d' - 1/2*B*e'` is symmetric positive
+definite with respect to the boundary quadrature. Here `d` is the normal
+derivative and `e` is the boundary restriction operator. `B` can then be used
+to form a symmetric and energy stable penalty for a Dirichlet condition. The
+parameters `H_tuning` and `R_tuning` are used to specify the strength of the
+penalty and must be greater than 1. For details we refer to
+<https://doi.org/10.1016/j.jcp.2020.109294>
+"""
+function positivity_decomposition(Δ::Laplace, g::Grid, b::BoundaryIdentifier; H_tuning, R_tuning)
+    @assert(H_tuning ≥ 1.)
+    @assert(R_tuning ≥ 1.)
+    Nτ_H, τ_R = positivity_limits(Δ,g,b)
+    return H_tuning*Nτ_H + R_tuning*τ_R
+end
+
+function positivity_limits(Δ::Laplace, g::EquidistantGrid, b::BoundaryIdentifier)
+    h = spacing(g)
+    θ_H = parse_scalar(Δ.stencil_set["H"]["closure"][1])
+    θ_R = parse_scalar(Δ.stencil_set["D2"]["positivity"]["theta_R"])
+
+    τ_H = one(eltype(Δ))/(h*θ_H)
+    τ_R = one(eltype(Δ))/(h*θ_R)
+    return τ_H, τ_R
+end
+
+function positivity_limits(Δ::Laplace, g::TensorGrid, b::BoundaryIdentifier)
+    τ_H, τ_R = positivity_limits(Δ, g.grids[grid_id(b)], b)
+    return τ_H*ndims(g), τ_R
+end