improve the code for boundary conditions

[Davide-Miotti]

Ensure that we have type-stable code for boundary types and implement a set of functions that leverage multiple dispatch when boundary conditions are enforced in the hermite scheme

[Davide-Miotti]

Now we have many if, elseif for example in _hermite_rbf_entry, this can be massively simplified using multiple dispatch.

if is_bound_i && !is_bound_j
        # Boundary-Interior: Apply boundary operator to first argument
        ni = data.normals[i]
        if is_dirichlet(bc_i)
            return φ
        else
            # Neumann/Robin: α*φ + β*∂ₙφ
            return α(bc_i) * φ + β(bc_i) * dot(ni, ∇φ)
        end

    elseif !is_bound_i && is_bound_j
        # Interior-Boundary: Apply boundary operator to second argument
        nj = data.normals[j]
        if is_dirichlet(bc_j)
            return φ
        else
            # Neumann/Robin: α*φ + β*∂ₙφ (note sign flip for gradient w.r.t. second arg)
            return α(bc_j) * φ + β(bc_j) * dot(nj, -∇φ)
        end

    else # is_bound_i && is_bound_j
        # Boundary-Boundary: Apply boundary operators to both arguments
        ni = data.normals[i]
        nj = data.normals[j]

        if is_dirichlet(bc_i) && is_dirichlet(bc_j)
            return φ
        elseif is_dirichlet(bc_i) && !is_dirichlet(bc_j)
            return α(bc_j) * φ + β(bc_j) * dot(nj, -∇φ)
        elseif !is_dirichlet(bc_i) && is_dirichlet(bc_j)
            return α(bc_i) * φ + β(bc_i) * dot(ni, ∇φ)
        else
            # Both Neumann/Robin: mixed derivative term
            ∂i∂j_φ = directional∂²(basis, ni, nj)(xi, xj)
            return (
                α(bc_i) * α(bc_j) * φ +
                α(bc_i) * β(bc_j) * dot(nj, -∇φ) +
                β(bc_i) * α(bc_j) * dot(ni, ∇φ) +
                β(bc_i) * β(bc_j) * ∂i∂j_φ
            )
        end
    end

[kylebeggs]

I think this is outdated now?

[Davide-Miotti]

Yes, we can close it