ISHIZAKA Hiroki, Ph. D. (Dr.)
石坂 宏樹
PhD-level Numerical Analyst｜Scientific‑Computing SME (PDEs & FEMs)｜AI Evaluator (Math/Reasoning/Numerical Methods)｜JP/EN
Self-introduction
ResearchGate Google Scholar researchmap Arxiv スライド保管 (Under construction)
"Mathematics for reliable simulation, built on geometry."
有限要素解の精度を上げる3つの戦略
"スキームを磨く" × "メッシュを活かす" × "両者の協調"
Research Attitude and Perspectives
幾何（メッシュ）が精度と安定性をどう支配するのか，という問いを出発点に，有限要素法（FEM）を「FEM 2.0」として再設計しています．FEM 2.0では，効率性（Efficiency），抽象化（Abstraction），自動化（Automation），応用展開（Application）という4つの目標を置き，私の研究はそれらを貫く3つの Research Line（G／R／A）に整理します．また，教育と研究の橋渡しとして，Supplementary Visionsで学びの地図を整理しています．
I study numerical analysis—the mathematics that makes computer simulations accurate and reliable.
My main tool is the finite element method (FEM), widely used to model physical systems such as fluids, heat, and materials. A central question guides my work:
- How does the geometry of the computational mesh control accuracy and stability?
- and how can we turn that knowledge into robust, partly automated FEMs?
- G - Geometry & Anisotropic FEM Theory
New geometric properties (flatness parameter, weakly regular meshes, two-step transformations) and anisotropic error estimates for nonstandard elements (CR, Morley, DG, Nitsche), including discrete Sobolev/Korn inequalities and mesh-geometry conditions. - R — Reliable & Goal-Oriented PDE Simulation
Time-dependent and memory-effect PDEs (BDF/FBDF, fractional-order in time), pressure-robust Stokes/Navier–Stokes, and goal-oriented / verified error control via functional and hypercircle-type estimators and equilibrated fluxes. - A — Automation, Learning & New Numerical Methods
Using geometric parameters and error indicators as features for adaptive FEM, mesh generation and scheme selection, and comparing FEM with PINNs/Deep Ritz/meshless methods under a common viewpoint of “representation – metric – projection”.
- Publications and Research show completed and near-completed results along these lines.
- Research Visions collects open directions and long-term ideas, indexed by the three lines above.
- History of Geometric Conditions in FEMs records the geometric side of Line G.
This site includes open directions (proposals). Published results are clearly labelled and linked on the Publications and Research pages.
"Research Visions"は日々増えてます．もっと整理した形で公開できるようにすることを検討中です．
左から，にこ淵・高樋沈下橋・佐田岬・ハイデルベルク大学図書館・ハイデルベルク大学数学・ハイデルベルク城・ドイツ バンメンタール
