--- a/diffOp.jl	Fri Jan 25 15:20:40 2019 +0100
+++ b/diffOp.jl	Thu Feb 21 16:27:28 2019 +0100
@@ -1,6 +1,7 @@
 abstract type DiffOp end
 
-function apply!(D::DiffOp, u::AbstractVector, v::AbstractVector)
+# TBD: The "error("not implemented")" thing seems to be hiding good error information. How to fix that? Different way of saying that these should be implemented?
+function apply(D::DiffOp, v::AbstractVector, i::Int)
     error("not implemented")
 end
 
@@ -32,50 +33,98 @@
     error("not implemented")
 end
 
-# Differential operator for a*d^2/dx^2
-struct Laplace{Dim,T<:Real,N,M,K} <: DiffOp
+abstract type DiffOpCartesian{Dim} <: DiffOp end
+
+# DiffOp must have a grid of dimension Dim!!!
+function apply!(D::DiffOpCartesian{Dim}, u::AbstractArray{T,Dim}, v::AbstractArray{T,Dim}) where {T,Dim}
+    for I ∈ eachindex(D.grid)
+        u[I] = apply(D, v, I)
+    end
+
+    return nothing
+end
+
+function apply_region!(D::DiffOpCartesian{2}, u::AbstractArray{T,2}, v::AbstractArray{T,2}) where T
+    apply_region!(D, u, v, Lower, Lower)
+    apply_region!(D, u, v, Lower, Interior)
+    apply_region!(D, u, v, Lower, Upper)
+    apply_region!(D, u, v, Interior, Lower)
+    apply_region!(D, u, v, Interior, Interior)
+    apply_region!(D, u, v, Interior, Upper)
+    apply_region!(D, u, v, Upper, Lower)
+    apply_region!(D, u, v, Upper, Interior)
+    apply_region!(D, u, v, Upper, Upper)
+    return nothing
+end
+
+# Maybe this should be split according to b3fbef345810 after all?! Seems like it makes performance more predictable
+function apply_region!(D::DiffOpCartesian{2}, u::AbstractArray{T,2}, v::AbstractArray{T,2}, r1::Type{<:Region}, r2::Type{<:Region}) where T
+    for I ∈ regionindices(D.grid.size, closureSize(D.op), (r1,r2))
+        @inbounds indextuple = (Index{r1}(I[1]), Index{r2}(I[2]))
+        @inbounds u[I] = apply(D, v, indextuple)
+    end
+    return nothing
+end
+
+function apply_tiled!(D::DiffOpCartesian{2}, u::AbstractArray{T,2}, v::AbstractArray{T,2}) where T
+    apply_region_tiled!(D, u, v, Lower, Lower)
+    apply_region_tiled!(D, u, v, Lower, Interior)
+    apply_region_tiled!(D, u, v, Lower, Upper)
+    apply_region_tiled!(D, u, v, Interior, Lower)
+    apply_region_tiled!(D, u, v, Interior, Interior)
+    apply_region_tiled!(D, u, v, Interior, Upper)
+    apply_region_tiled!(D, u, v, Upper, Lower)
+    apply_region_tiled!(D, u, v, Upper, Interior)
+    apply_region_tiled!(D, u, v, Upper, Upper)
+    return nothing
+end
+
+using TiledIteration
+function apply_region_tiled!(D::DiffOpCartesian{2}, u::AbstractArray{T,2}, v::AbstractArray{T,2}, r1::Type{<:Region}, r2::Type{<:Region}) where T
+    ri = regionindices(D.grid.size, closureSize(D.op), (r1,r2))
+    for tileaxs ∈ TileIterator(axes(ri), padded_tilesize(T, (5,5), 2)) # TBD: Is this the right way, the right size?
+        for j ∈ tileaxs[2], i ∈ tileaxs[1]
+            I = ri[i,j]
+            u[i,j] = apply(D, v, (Index{r1}(I[1]), Index{r2}(I[2])))
+        end
+    end
+    return nothing
+end
+
+function apply(D::DiffOp, v::AbstractVector)::AbstractVector
+    u = zeros(eltype(v), size(v))
+    apply!(D,v,u)
+    return u
+end
+
+struct Laplace{Dim,T<:Real,N,M,K} <: DiffOpCartesian{Dim}
     grid::Grid.EquidistantGrid{Dim,T}
     a::T
     op::D2{Float64,N,M,K}
 end
 
-# u = L*v
-function apply!(L::Laplace{1}, u::AbstractVector, v::AbstractVector)
-    h = Grid.spacings(L.grid)[1]
-    apply!(L.op, u, v, h)
-    u .= L.a * u
-    return nothing
+function apply(L::Laplace{Dim}, v::AbstractArray{T,Dim} where T, I::CartesianIndex{Dim}) where Dim
+    error("not implemented")
 end
 
 # u = L*v
-function apply!(L::Laplace{2}, u::AbstractVector, v::AbstractVector)
-    u .= 0*u
-    h = Grid.spacings(L.grid)
-
-    li = LinearIndices(L.grid.numberOfPointsPerDim)
-    n_x, n_y = L.grid.numberOfPointsPerDim
-
-
-    # For each x
-    temp = zeros(eltype(u), n_y)
-    for i ∈ 1:n_x
-
-        v_i = view(v, li[i,:])
-        apply!(L.op, temp, v_i, h[2])
+function apply(L::Laplace{1}, v::AbstractVector, i::Int)
+    uᵢ = L.a * apply(L.op, L.grid.spacing[1], v, i)
+    return uᵢ
+end
 
-        u[li[i,:]] += temp
-    end
+@inline function apply(L::Laplace{2}, v::AbstractArray{T,2} where T, I::Tuple{Index{R1}, Index{R2}}) where {R1, R2}
+    # 2nd x-derivative
+    @inbounds vx = view(v, :, Int(I[2]))
+    @inbounds uᵢ = L.a*apply(L.op, L.grid.inverse_spacing[1], vx , I[1])
+    # 2nd y-derivative
+    @inbounds vy = view(v, Int(I[1]), :)
+    @inbounds uᵢ += L.a*apply(L.op, L.grid.inverse_spacing[2], vy, I[2])
+    return uᵢ
+end
 
-    # For each y
-    temp = zeros(eltype(u), n_x)
-    for i ∈ 1:n_y
-        v_i = view(v, li[:,i])
-        apply!(L.op, temp, v_i, h[1])
-
-        u[li[:,i]] += temp
-    end
-
-    u .= L.a*u
-
-    return nothing
+# Slow but maybe convenient?
+function apply(L::Laplace{2}, v::AbstractArray{T,2} where T, i::CartesianIndex{2})
+    I = Index{Unknown}.(Tuple(i))
+    apply(L, v, I)
 end