My doctoral research involved developing a numerical algorithm to efficiently solve the spectral fractional Laplacian. We can dissect this topic starting with the Laplacian equation itself. Recall that the Laplacian is \[ (-\Delta)u = f \,\, , \,\, x \in \Omega \, . \] We will restrict ourselves to the case where \(u=0 \,\, , \,\, x \in \partial \Omega\,\). The Laplacian is a second-order elliptic partial differential equation and represents diffusion, or spreading out, of some quantity \(u(x)\). The modifier fractional to the Laplacian means that we apply some fraction of the Laplacian operator to that quantity \(u(x)\) and can be written as \[ (-\Delta)^s = f \,\, , \,\, x \in \Omega \, , \] where \(s \in (0,1)\). The fractional Laplacian can be defined in numerous ways. See this paper or this paper to see some the definitions.

The spectral fractional Laplacian is a particular interpretation. To get a better idea we will introduce a results from spectral theory. Spectral theory tells us that \(-\Delta\) has a countable set of eigenpairs \(\{ \lambda_k, \phi_k \}_k\) such that the \(\phi_k\)’s form an orthonormal basis of \(L^2(\Omega)\). So if we have some smooth function \(w(x)\), we can expand this function in that basis, i.e., \[ w(x) = \sum_k w_k(x) \phi_k(x) \,\, , \] where \(w_k\) is the Fourier coefficient of \(w(x)\) with respect to the k-th eigenfunction, \(\phi_k(x)\). Applying the Laplacian, \[ (-\Delta) w(x) = \sum_k w_k (-\Delta)\phi_k(x) = \sum_k w_k \lambda_k \phi_k(x) \,\, . \] Now, using this we can define the spectral fractional Laplacian as: \[ (-\Delta)^s w(x) = \sum_k w_k (-\Delta)^s \phi_k(x) = \sum_k w_k \lambda_k^s \phi_k(x) \,\, . \]

The takeaway is that if we know the eigenpairs we can write the solution to the spectral fractional Laplacian directly. Unfortunately, finding the eigenpairs is both computationally expensive and unstable. Fortunately, Caffarelli and Silvestre showed that the problem of the spectral fractional Laplacian in \(d\) dimensions is related to a problem in \(d+1\) dimensions with no fractional derivatives. This extended problem is amenable to solving by the finite element method and was extensively studied by doctoral advisor Dr. Abner Salgado and his collaborators. A major result from this paper is that the \(d+1\) dimensional problem can be decomposed, separating the problem in \(\Omega\) from the extended dimension. The problem in the extended dimension can be further simplified by introducing an auxiliary eigenvalue problem. Working with my advisor, I found the analytical solution to the auxiliary problem rather than numerically solve the disretized eigenvalue problem as has been previously done.

Perhaps the most significant contribution of my research came about during the error analysis of the new method. In showing the theoretical convergence rate, I was able to show that the new method is a quadrature for the Balakrishann formulation of the spectral fractional Laplacian. This hints at a deeper connection between the interpretations of the fractional Laplacian.

Having a computational and scientific programming background, I implemented the method in C++ using the deal.ii finite element library. I was able to rapidly implement my algorithm and demonstrate that it recovers the theoretical convergence rate in practice. I was similarly able to show the algorithm was efficient in the sense that it exhibits excellent strong and weak parallel scaling.

Links

• Dissertation Defense Slides
• Access my dissertation here - On embargo until 8/15/2025
• Submitted Manuscript
• deal.ii candi Repository - Recommended for building deal.ii with libraries on Linux
• Source Code Repository