# Boundary conditions for an FEM approximation of the Laplace operator

Using FEM, I want to approximate the Laplacian

$$u = \nabla \cdot \nabla h \, ,$$

where $$h(x,y)$$ is an FEM approximated scalar field on the same mesh, i.e. piecewise differentiable.

I am using MOOSE to solve the following (hopefully correct!) weak formulation of the above equation:

$$\int_\Omega uw_i \, \text{d}\Omega = -\int_\Omega \nabla h \cdot \nabla w_i \, \text{d}\Omega + \int_\Gamma \nabla h \cdot \vec{n}w_i \, \text{d}\Gamma \,,$$

where $$w_i$$ is the i-th test function and $$\vec{n}$$ is the normal vector on the boundary $$\Gamma$$ of the computational domain $$\Omega$$. I am not using any boundary conditions on $$u$$.

The FEM solution that I obtain for this weak formulation is inaccurate at the boundaries (see below). I have tested this by prescribing the field $$h=\text{sin}(x)$$ on a 2D rectangular domain from $$-2\pi \le x \le 2\pi$$ and solving with quadratic Lagrange shape functions for both $$h$$ and $$u$$. (it is also inaccurate using linear shape functions, but it is easier to spot with quadratic shape functions).

Is it possible to obtain an accurate $$u$$ at the boundaries with this approach? Am I missing a boundary condition/contribution?

Background:

I am dealing with a thin film approximation of the Navier-Stokes equations in 2D FEM, where the film height is modeled by a dependent variable $$h(x,y)$$. In order to include surface tension as a contribution to the pressure gradient, I want to find the curvature of the film surface $$u(x,y)$$ that I approximate by the second derivative

$$u \approx \nabla \cdot \nabla h$$ .

My idea is to introduce curvature $$u$$ as a variable, solve the above equation by FEM, and couple $$u$$ into the momentum equation.

• Let me see if I understand what you want to achieve. You have a function $h(x, y)$ and you want to compute it's Laplacian over a domain? Mar 29, 2021 at 18:04
• Are you ignoring the boundary term from your weak form? The formulation you have should show up with a Neumann boundary term for GFEM.
– wwfe
Mar 29, 2021 at 18:11
• @nicoguaro Yes, but let me reformulate: I want to calculate u by FEM so that it approximates the Laplacian of h(x, y) ("source term" on the right hand side of the equation). I don't have a closed form of h(x, y) though, because it is actually a variable in another FEM problem. Mar 29, 2021 at 23:10
• @wwfe Can you tell me what boundary term you mean and what it should look like? I don't have a boundary term with u since I am not integrating by parts on the LHS of the pde. Mar 29, 2021 at 23:13
• @pimpom the boundary represented by the integral over $\Gamma$ is in your equation, just because u isn't there doesn't mean it isn't a boundary term, it is still part of the residual.
– wwfe
Mar 30, 2021 at 3:04