Book ChapterDOI
On Computability of Navier-Stokes’ Equation
Shu Ming Sun,Ning Zhong,Martin Ziegler +2 more
- pp 334-342
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TLDR
A suitable encoding is carefully constructed for the space of solenoidal vector fields in the \(L_q\) sense over the \(d\)-dimensional open unit cube with zero boundary condition to render both the Helmholtz projection and the semigroup generated by the Stokes operator uniformly computable in the case of q=2.Citations
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Journal ArticleDOI
On the computational complexity of the Dirichlet Problem for Poisson's Equation
TL;DR: It is established that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense ‘complete’ for the complexity class P and thus as hard or easy as parametric Riemann integration.
Proceedings ArticleDOI
Bounded time computation on metric spaces and banach spaces
TL;DR: In this article, the authors extend Kawamura and Cook's framework for computational complexity for operators in analysis to metric spaces via representations, where time is measured in the length of the input encodings and the output precision.
Book ChapterDOI
Towards Computational Complexity Theory on Advanced Function Spaces in Analysis
TL;DR: The guide is relativization: Permitting arbitrary oracles on continuous universes reduces computability to topology and computational complexity to metric entropy in the sense of Kolmogorov.
Book ChapterDOI
Computability of the Solutions to Navier-Stokes Equations via Effective Approximation
TL;DR: The question of whether the Navier-Stokes Equation admits recursive solutions in the sense of Weihrauch’s Type-2 Theory of Effectivity is approached and a natural encoding is constructed for the space of divergence-free vector fields on 2-dimensional open square.
Posted Content
Computability of the Solutions to Navier-Stokes Equations via Recursive Approximation
TL;DR: This paper approaches the question of whether the Navier-Stokes Equation admits recursive solutions in the sense of Weihrauch's Type-2 Theory of Effectivity and establishes the computability results.
References
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Book
Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms
TL;DR: This paper presents the results of an analysis of the "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions and its applications to Mixed Approximation and Homogeneous Stokes Equations.
Book
Computability in analysis and physics
TL;DR: This book represents the first treatment of computable analysis at the graduate level within the tradition of classical mathematical reasoning and is sufficiently detailed to provide an introduction to research in this area.
Journal ArticleDOI
Solutions in Lr of the Navier-Stokes initial value problem
Yoshikazu Giga,Tetsuro Miyakawa +1 more
Journal ArticleDOI
The wave equation with computable initial data such that its unique solution is not computable
TL;DR: In this paper, it was shown that all non-computable solutions of the wave equation are of the type commonly referred to as weak solutions, i.e., although continuous, they are not twice differentiable at all points.
Journal ArticleDOI
Computability and Recursion
TL;DR: After a careful historical and conceptual analysis of computability and recursion, several recommendations are made about preserving the intensional differences between the concepts of “computability” and “recursion.”
Related Papers (5)
Time-Periodic Linearized Navier–Stokes Equations: An Approach Based on Fourier Multipliers
Thomas Eiter,Mads Kyed +1 more