A coercive space-time variational approach to fractional diffusion problems

Abstract

We consider a fractional diffusion problem with temporal nonlocality acting on the diffusive flux. A coercive space--time variational formulation in Bochner-valued fractional Sobolev spaces is derived and the existence, uniqueness, and regularity of solutions are established. We further develop a conforming tensor-product Galerkin discretization and prove quasi-optimal error estimates in the anisotropic energy norm and improved convergence rates in weaker norms using duality arguments. In contrast to some space-time formulations for classical diffusion, the method preserves the causal structure of the evolution problem and leads to a time-stepping procedure with memory terms. On uniform time grids, the discrete history operator has a lower-triangular Toeplitz structure which enables an efficient implementation using fast recursive convolution techniques.

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BibTeX

@misc{egger2026coercive,
  title = {{A coercive space-time variational approach to fractional diffusion problems}},
  author = {Herbert Egger and Marvin Fritz and Barbara Wohlmuth},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00091},
  eprint = {2609.00091},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00091},
}

RIS

TY  - UNPB
TI  - A coercive space-time variational approach to fractional diffusion problems
AU  - Herbert Egger
AU  - Marvin Fritz
AU  - Barbara Wohlmuth
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00091
UR  - https://arxiv.org/abs/2609.00091
AB  - We consider a fractional diffusion problem with temporal nonlocality acting on the diffusive flux. A coercive space--time variational formulation in Bochner-valued fractional Sobolev spaces is derived and the existence, uniqueness, and regularity of solutions are established. We further develop a conforming tensor-product Galerkin discretization and prove quasi-optimal error estimates in the anisotropic energy norm and improved convergence rates in weaker norms using duality arguments. In contrast to some space-time formulations for classical diffusion, the method preserves the causal structure of the evolution problem and leads to a time-stepping procedure with memory terms. On uniform time grids, the discrete history operator has a lower-triangular Toeplitz structure which enables an efficient implementation using fast recursive convolution techniques.
ER  -

Abstract

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