Alphabeta MathNumerical Pdes and Spatial Discretization
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
12 pagesThis page has two independent branches: a selected one-dimensional moving-mesh finite-volume scheme, and a finite-dimensional linear diffusion-transmission problem.
4 definitions, 2 algorithms, 4 lemmas, 6 theorems, 1 corollaryExamples & counterexamples →This page defines a one-dimensional full-line conservative finite-volume scheme for a scalar Cauchy problem, its cellwise state and face-flux reconstructions, and the Godunov flux for Burgers' equation.
4 definitions, 3 lemmas, 1 theoremExamples & counterexamples →- Elliptic finite-difference methods4 results
On a rectangular uniform grid, this page defines the five-point Dirichlet Laplacian and a scalar divergence-form flux stencil with midpoint face coefficients.
1 definition, 2 lemmas, 1 theoremExamples & counterexamples → - Finite-element meshes and interpolation24 results
A finite face-to-face affine simplicial mesh is the setting for the interpolation and differential identities on this page.
5 definitions, 9 lemmas, 10 theoremsExamples & counterexamples → This page separates nodal ENO2 from finite-volume cell-average reconstruction, defines minmod MUSCL, WENO5-JS and the WENO-Z comparison, and proves their stated smooth-region face accuracy.
1 definition, 2 lemmas, 4 theorems, 1 counterexampleExamples & counterexamples →The mean-field model uses a globally Lipschitz two-variable interaction kernel and probability measures with finite first moment.
9 definitions, 1 algorithm, 5 lemmas, 13 theoremsExamples & counterexamples →This page defines a common Banach-space interface for a well-posed autonomous linear initial-value problem and its time-stepping family.
2 definitions, 1 lemma, 2 theoremsExamples & counterexamples →This page proves kernel smoothing and particle-cell quadrature bounds for a symmetric normalized compactly supported kernel.
3 definitions, 6 lemmas, 2 propositions, 5 theorems, 1 corollaryExamples & counterexamples →- Nonlinear full-approximation multigrid11 results
Full-approximation scheme (FAS) applies multigrid directly to nonlinear residual equations.
2 algorithms, 6 lemmas, 2 theorems, 1 exampleExamples & counterexamples → This page sets up finite-rank Nyström operators and proves their equivalence with the sampled grid equations.
2 definitions, 7 lemmas, 4 theoremsExamples & counterexamples →- Riemann solvers for hyperbolic systems25 results
This page develops the local theory and the corresponding first-order fluxes for one-dimensional strictly hyperbolic conservation systems.
10 definitions, 1 algorithm, 8 lemmas, 6 theoremsExamples & counterexamples → This page defines periodic Fourier and Chebyshev–Gauss–Lobatto collocation, including the even-grid Nyquist convention, exact differentiation of the chosen interpolants, and…
7 definitions, 5 algorithms, 4 lemmas, 12 theorems, 1 remarkExamples & counterexamples →
Level 1
9 pagesFor a common bounded coercive form and a conforming finite-dimensional subspace, Lax–Milgram gives the continuous and discrete solutions, Galerkin orthogonality gives Céa's…
2 lemmas, 3 theorems, 4 corollariesExamples & counterexamples →This draft develops the real Stokes mixed formulation, a bounded divergence right inverse on bounded connected Lipschitz domains, the Babuška–Brezzi kernel-coercivity…
5 definitions, 11 lemmas, 10 theoremsExamples & counterexamples →This page proves strong local entropy convergence for first-order scalar finite-volume schemes with a locally Lipschitz, consistent, monotone two-point flux.
3 definitions, 3 lemmas, 1 theoremExamples & counterexamples →- Parabolic finite-difference methods9 results
This page defines the fully discrete centered heat θ update on periodic grids and rectangular Dirichlet grids, including initial data and exact boundary rows.
1 definition, 3 lemmas, 5 theoremsExamples & counterexamples → This page fixes a periodic electrostatic particle/grid model on T Lᵈ, d∈{1,2,3}, with an odd uniform grid, a nonnegative partition-of-unity deposition shape, a uniform…
4 definitions, 2 algorithms, 3 lemmas, 14 theoremsExamples & counterexamples →This page gives a first-order HLL invariant-domain result from the exact two-wave flux algebra.
6 definitions, 1 algorithm, 2 lemmas, 1 proposition, 4 theoremsExamples & counterexamples →- Substructuring, FETI-DP and BDDC19 results
This page develops the primal and dual interface formulations for a conforming scalar P 1 discretization of second-order diffusion.
5 definitions, 5 lemmas, 2 propositions, 6 theorems, 1 remarkExamples & counterexamples → This page defines the symmetric interior-penalty method for scalar, real, second-order elliptic diffusion on finite matching shape-regular affine simplicial meshes, with a…
2 definitions, 5 lemmas, 4 theoremsExamples & counterexamples →This page defines the three-dimensional outgoing volume integral model on L²(D), proves boundedness and compactness with the singular diagonal treated explicitly, and…
3 definitions, 7 lemmas, 6 theoremsExamples & counterexamples →
Level 2
12 pages- Adaptive finite-element methods20 results
The prerequisite page variational-crimes-and-strang-lemmas supplies the conforming variational framework.
4 definitions, 1 algorithm, 10 lemmas, 4 theorems, 1 corollaryExamples & counterexamples → - Finite-element eigenvalue approximation10 results
For a bounded Hermitian coercive form with compact, injective, dense mass embedding, the page establishes the compact solution-operator spectrum and the attained continuous min–max principle under Countable Choice.
1 definition, 4 lemmas, 1 proposition, 4 theoremsExamples & counterexamples → This page defines the upwind DG method for constant scalar transport on one-dimensional interval meshes and on matching affine simplicial meshes in dimensions at least two.
6 definitions, 1 algorithm, 1 lemma, 6 theoremsExamples & counterexamples →- Hyperbolic finite-difference methods11 results
This page defines five scalar advection stencils on periodic or infinite uniform grids and the upwind inflow row on a finite interval.
3 definitions, 3 lemmas, 5 theoremsExamples & counterexamples → For bounded connected Lipschitz domains in dimensions two and three, this page defines the homogeneous-Dirichlet small-strain isotropic elasticity model with measurable shear modulus 0<μ 0≤μ≤μ 1 and finite Lamé parameter λ≥0.
2 definitions, 1 algorithm, 2 lemmas, 2 theoremsExamples & counterexamples →This page develops real H(div) fluxes with the Green-duality normal trace, then defines Raviart–Thomas and Brezzi–Douglas–Marini polynomial families on conforming affine simplices.
7 definitions, 14 lemmas, 6 theoremsExamples & counterexamples →Nitsche's method imposes Dirichlet data in an H¹-conforming finite-element space without restricting the trial functions to have the prescribed trace.
2 definitions, 8 lemmas, 4 theoremsExamples & counterexamples →This page defines exact and spectrally equivalent inexact Schwarz local solves for finite-dimensional SPD diffusion energies.
1 definition, 4 lemmas, 4 theoremsExamples & counterexamples →- Parabolic finite-element methods21 results
The page treats parabolic evolution in a real Hilbert triple, first proving the time-trace and energy identities needed for variational well-posedness.
2 definitions, 2 algorithms, 10 lemmas, 6 theorems, 1 corollaryExamples & counterexamples → The projection results here concern the unit torus Tᵈ, a uniform periodic grid with N≥2, face-centred velocity arrays, cell-centred pressure arrays, and the mean-zero pressure gauge.
2 definitions, 2 algorithms, 2 lemmas, 7 theorems, 2 corollariesExamples & counterexamples →- Variational crimes and Strang lemmas12 results
The first Strang estimate treats conforming Galerkin spaces with perturbed forms and loads.
2 definitions, 5 lemmas, 2 theorems, 3 corollariesExamples & counterexamples → This page defines the one-dimensional shallow-water balance law with a dry-state convention and the first-order hydrostatic HLL reconstruction.
2 definitions, 6 lemmas, 1 proposition, 3 theoremsExamples & counterexamples →
Level 3
9 pages- Crouzeix–Raviart finite elements20 results
The P1 Crouzeix–Raviart space consists of cellwise affine functions whose facet means agree on interior facets and vanish on boundary facets.
6 definitions, 10 lemmas, 4 theoremsExamples & counterexamples → This page defines admissible isoparametric maps and geometry-approximation errors, proves pullback and interpolation estimates, and bounds the Laplace tensor defect used in a local Strang estimate.
3 definitions, 2 algorithms, 8 lemmas, 5 theoremsExamples & counterexamples →On an oriented matching tetrahedral mesh, the local edge field associated with an oriented edge [i,j] is φ ij=λ i∇λ j-λ j∇λ i.
4 definitions, 15 lemmas, 5 theorems, 1 counterexampleExamples & counterexamples →This page fixes the PE30 plan-spec sign convention for the first-order Poisson system: σ=-∇ u, divσ=f, and residual (σ+∇ u,divσ-f).
2 definitions, 2 algorithms, 9 lemmas, 5 theorems, 1 corollary, 1 remarkExamples & counterexamples →This page develops finite weighted graph diffusion and diffusion on finite metric graphs.
4 definitions, 5 lemmas, 5 theorems, 1 corollaryExamples & counterexamples →- hp finite elements and spectral elements22 results
This page defines variable-degree spaces on matching affine simplicial and tensor-box meshes.
3 definitions, 9 lemmas, 10 theoremsExamples & counterexamples → This page proves continuous and piecewise-constant discrete Aubin–Lions compactness, then uses nonlinear backward Euler Galerkin steps, energy bounds, weak space-time…
3 definitions, 9 lemmas, 7 theoremsExamples & counterexamples →This page keeps the homogeneous stationary one-dimensional P1 branch and its exact cell parameter τ=h/2|b|(coth(Pe)-Pe⁻¹) separate from a multidimensional P1 branch with c-1/2∇· b≥ω 0 and an explicit small-parameter condition.
6 definitions, 3 lemmas, 5 theoremsExamples & counterexamples →This page develops frictionless unilateral contact for small-strain isotropic elasticity on a bounded connected planar Lipschitz polygon assumed to be a W^1,2-extension domain.
5 definitions, 8 lemmas, 3 theoremsExamples & counterexamples →
Level 4
6 pagesThis draft treats Newtonian incompressible Navier–Stokes on bounded connected Lipschitz polygonal or polyhedral domains in dimensions two and three, with viscosity ν 0…
4 definitions, 12 lemmas, 6 theorems, 1 remarkExamples & counterexamples →This page defines finite-volume cell-face meshes, isotropic two-point flux approximation (TPFA) on orthogonal center-to-face geometries, and a hybrid mimetic method for anisotropic diffusion on general cells.
4 definitions, 3 lemmas, 1 proposition, 5 theoremsExamples & counterexamples →- Isogeometric Galerkin discretization19 results
This page defines tensor-product NURBS geometry with positive bounded weights, globally injective and bi-Lipschitz patch maps, positive cell Jacobians, and finite conforming…
2 definitions, 9 lemmas, 8 theoremsExamples & counterexamples → The page formulates the PEC curl-curl eigenproblem on the zero-tangential H(curl) space.
1 definition, 7 lemmas, 1 theoremExamples & counterexamples →- Semi-Lagrangian characteristic transport12 results
This page develops periodic semi-Lagrangian transport for the scalar equation u t+b·∇ u=0.
2 definitions, 1 algorithm, 6 lemmas, 3 theoremsExamples & counterexamples → This page defines real transient Maxwell fields with uniformly positive material tensors, source continuity, periodic or PEC boundaries, and transmission conventions for aligned material interfaces.
4 definitions, 1 algorithm, 8 lemmas, 7 theoremsExamples & counterexamples →
Level 5
5 pagesThe page defines the positive ideal-gas state domain, derives the strictly convex mathematical entropy and its entropy variables, and factors the Newtonian–Fourier viscous operator into symmetric positive-semidefinite blocks.
4 definitions, 1 algorithm, 7 lemmas, 3 theoremsExamples & counterexamples →This page treats measurable, uniformly elliptic scalar coefficients that are periodic on the unit lattice.
5 definitions, 1 algorithm, 10 lemmas, 5 theorems, 1 corollaryExamples & counterexamples →This page develops three related finite-control HJB branches on the periodic unit torus and a self-contained second-order comparison chain.
6 definitions, 4 algorithms, 12 lemmas, 11 theorems, 1 corollaryExamples & counterexamples →This page fixes the quartic double-well energy and the no-flux Allen–Cahn (AC) and Cahn–Hilliard (CH) gradient flows in Allen–Cahn and Cahn–Hilliard double-well gradient flows.
3 definitions, 2 algorithms, 2 lemmas, 4 theoremsExamples & counterexamples →This page defines locally Lipschitz reaction fields, inward invariant rectangles, stoichiometric conservation and dissipation, continuous semilinear systems, and the periodic…
7 definitions, 2 algorithms, 3 lemmas, 1 proposition, 7 theoremsExamples & counterexamples →
Level 6
1 pageThis page develops three-dimensional Laplace boundary integral equations on a bounded connected domain with a closed, connected, orientable Lipschitz polyhedral boundary and connected exterior.
9 definitions, 1 algorithm, 17 lemmas, 11 theorems, 1 corollaryExamples & counterexamples →
Level 7
1 pageThe exterior model and angular, radial-flux and outgoing-mode foundations are placed on boundary-integral-equations-and-galerkin-bem, retaining their original item identifiers.
4 definitions, 2 lemmas, 7 theoremsExamples & counterexamples →