| Abstract: |
| The $H^\infty$-calculus provides a powerful framework for establishing well-posedness and optimal regularity results for stochastic partial differential equations (SPDEs). In particular, it is the cornerstone for proving stochastic maximal regularity, a crucial tool for handling non-linear SPDEs via linearization techniques. While this continuous theory is well-established, its applications to numerical analysis---specifically for deriving sharp stability and convergence rates for numerical schemes---have only recently begun to emerge. In this talk, I will present a framework for establishing a priori regularity estimates for semi-discrete numerical schemes of linear SPDEs, which subsequently enable the convergence analysis of non-linear problems. The core of this approach relies on proving the uniform boundedness of the $H^\infty$-calculus for the finite element discretizations $\{A_h \colon h>0\}$ of a second-order elliptic operator $A$. |
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