At the Scientific Board Meeting on 24 June 2026, the FWF decided to fund the individual research project “Computational Fractional Diffusion” (project number PAT1917426, opens an external URL in a new window), led by Ass.Prof. Markus Faustmann, with approximately €441,000.
The project focuses on the numerical treatment of fractional differential equations. Such equations often describe dynamical processes involving nonlocal or memory effects. Applications can be found in fields including physics, biology, image processing, and finance. Since the underlying mathematical models generally cannot be solved exactly, powerful numerical methods are needed to approximate their solutions and simulate the models on a computer.
The project will develop and analyze finite element methods (FEM) for the approximate solution of specific classes of fractional differential equations. The research will investigate, among other topics, operators with nonlocal Neumann boundary conditions that model flux processes out of an object, as well as operators with variable fractional exponents that can be used, for example, in image processing. In addition, the project will address physically relevant nonlinear equations, such as fractional Schrödinger equations and peridynamic equations.
A particular mathematical challenge is that solutions of such equations generally exhibit strong singularities, which can affect the reliability and accuracy of numerical computations. At the same time, the nonlocal nature of fractional differential operators can potentially lead to high memory requirements and long computation times. The project will address these challenges on two levels. On the one hand, a priori error estimates for FEM approximations on local subdomains will be developed, and the regularity and decay behaviour of solutions will be rigorously investigated. On the other hand, efficient methods such as hp-FEM, adaptive finite element methods, and matrix compression techniques will be analyzed for the model problems under consideration, with the aim of enabling the efficient numerical solution of these equations on a computer.