Adaptive Quarklet Methods for the Numerical Solution of Elliptic Partial Differential Equations with Exponential Convergence

• Applicant: Dr. Marc Hovemann
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 528343051

This project is concerned with the development of efficient next generation adaptive numerical methods for the solution of partial differential equations using the recently introduced Quarklets with provable exponential convergence. For that purpose we work with biorthogonal compactly supported Cohen-Daubechies-Feauveau spline wavelets, that are enriched with polynomials of certain degree, namely the so-called Quarklets. They allow for the construction of wavelet counterparts of adaptive hp-finite element methods. To develop effective Quarklet methods with exponential convergence in a first step we have to assemble multivariate Quarklet systems with high smoothness, that can be used to characterize advanced function spaces such as Besov- or Triebel-Lizorkin spaces. Those multivariate Quarklets will be used to design adaptive near-best Quarklet tree approximation techniques for given multivariate functions. For such methods we want to identify associated approximation classes, namely a large class of multivariate functions, for that our adaptive near-best Quarklet tree approximation procedure converges with exponential order. In a next step an efficient next generation adaptive Quarklet PDE solver will be developed, that is based on a damped Richardson iteration, and uses our adaptive near-best Quarklet tree approximation technique as an important building block. For our Quarklet PDE solver then situations (in terms of conditions on the PDE, the underlying domain and the right-hand side) should be identified, such that exponential convergence can be proved. For that purpose several techniques out of regularity theory for PDEs will be used. Along with the project goes the practical implementation of all numerical Quarklet methods we designed.

• DFG Programme: Research Grants