Math

• Solution uniqueness and stability of the variational problems for elliptic PDEs

• Understanding about the GMRES method

• Conclusions related to special matrix patterns

• Minimization of a functional for solving linear problems

• Understanding about Jacobian matrix

• Understanding about the Lagrange multiplier method and its application in PDEs

• Two kinds of projections in solving linear problems

• General theory about the construction of preconditioning bilinear form in BEM

• Moore-Penrose pseudoinverse and generalized inverse

• Ellipticity of boundary integral operators

• Basic ideas of operator preconditioning

• Adjoint operators in functional analysis

• Spectral equivalence

• Kernel and range of a matrix and its transpose

• Variational problems

• Two kinds of coordinate transformations in differential geometry

• Riemannian metric for a submanifold within \(\mathbb{R}^N\)

• Integration of differential forms and its relationship with classical calculus

• Understanding about the orientation dependence of pseudo differential forms

• The interplay between triangulation, manifold, mapping, quadrature point and finite element

• Concept of duality

• Shape functions and basis functions in FEM

• Summary of Green's identities

• Rumination on first order electrical circuit

• Understanding about operator discretization

• From duality pairing to operator preconditioning

• Understanding about Riesz map

• Domain, range and dual spaces in BEM

• 关于数值优化的定性理解

• Understanding about energy norm used in Galerkin method

• Fundamental theorems in PDE theory

• Understanding about isomorphism

• Different types of convergence

• Understanding about ellipticity of operators

• Schur complement

• 关于行列式的理解

• The relation between Riemannian metric, area form and complex structure

• Formulation of complex structure

• Visualization of Hilbert curve in Asymptote

• Understanding about abstract Riemannian metric

• Understanding about representing regular spatial curve using differential form

• Algorithm for reconstruction of a discrete space curve

• Understanding about the coboundary of coboundary is empty

• Understanding about the anti-symmetry of \(k \)-forms and geometric meaning of determinant

• Understanding about integration

• The meaning of applying a covector to a vector

• Understanding about the sharp and flat operators

• Measurement and duality

• The meaning of Jacobian matrix having a rank \(r\)

• The star of a subset of simplices

• Construct topological data structures from a mesh

• Voltage distribution simulation using 3D Galerkin BEM

• 边界元算法开发状态与算法库简介

• Clarification of basis, base and bases

• Typical equivalence relations between mathematical spaces

• Understanding about submanifold and manifold

• Verification of singularity order of BEM kernels

• Summary of Sobolev spaces and their norms

• Matrix assembly paradigms in BEM

• Discretization of bilinear forms in BEM

• Difference between interpolation and projection in FEM

• 聚合物绝缘空间电荷动力学的理论模型与数值仿真研究进展

• Prove a function space is a Banach space

• Summary of multi-index convention for partial derivatives

• Isometric embedding of metric space

• Mindmap for "Principles of Boundary Element Methods"

• Exercise 6 in Section 22 of James Munkres Topology

• Metaphor of quotient space

• Exercise 3 in Section 22 of James Munkres Topology

• Example 1 in Section 22 of James Munkres Topology

• Continuity of arithmetic operations

• The sequence lemma in James Munkres Topology

• Matrix norm

• Theorem 20.3 and metric equivalence in James Munkres Topology

• Theorem 20.4 in James Munkres Topology

• Theorem 19.6 in James Munkres Topology

• A tuple is defined as a function

• Different ways of constructing continuous functions

• Exercise 12 in Section 18 of James Munkres Topology

• Understanding of continuity definition in topology

• Theorem 16.3 in James Munkres Topology

• Barber paradox

• Metaphor of topological basis and open set