2021/04/19 by Frank Rösler, Rösler, Frank, Alexei Stepanenko +1
Computer Science · Mathematics · #35P15 #65N25 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical Analysis (math.NA) #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2104.09444
openalex publication_date 2021/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counter example showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.