Galerkin method

Defining discrete condition

For the conforming linear formulation, seek u_h in V_h with a(u_h,v_h)=l(v_h) for every v_h in V_h. [Massing's Galerkin course notes](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_02/galerkin_methods.html).

1. What residual functional must vanish, and against which complete discrete test space? definition
2. Does testing use an inner product, a duality pairing or a specified discrete approximation? definition

Named variant contract

Record what each variant label changes in the spaces, differential operators or discrete form.

1. Does the cited author use Galerkin narrowly for equal variation spaces or broadly to include Petrov-Galerkin formulations? provenance
2. What equations establish the claimed relationship to Ritz, collocation or least-squares methods? boundary
3. Which features belong to a linked specialised variant rather than to every Galerkin application? boundary

Formulation equivalence

Record the operator, forcing, domain, regularity requirements and derivation of the tested formulation.

1. What operator equation or variational problem is being approximated? definition
2. Under which regularity assumptions does the chosen weak formulation represent the intended original problem? boundary

Constraint realisation

Track essential and natural boundary conditions, lifting functions, compatibility conditions and any nullspace constraints.

1. Which boundary conditions are imposed through spaces, lifting, boundary terms or weak enforcement? definition
2. What compatibility or normalisation condition is required if the operator has a nullspace? action

Space and basis record

Record basis construction, dimensions, support, continuity and component structure.

1. How are trial and test spaces constructed, and how many independent degrees of freedom does each contain? definition
2. Are apparently different bases spanning the same space or genuinely different spaces? boundary

Admissible refinement

Record continuous-space membership, interface treatment and how a sequence of spaces gains approximation power.

1. Do the discrete functions satisfy the continuous space requirements, or which broken-space or nonconforming construction replaces them? boundary
2. Should accuracy be improved by mesh refinement, degree increase, additional modes or basis enrichment? action
3. What approximation or density evidence supports the selected refinement sequence? provenance

Coefficient system

For a linear form, basis substitution gives A_ij=a(phi_j,phi_i) and b_i=l(phi_i). [Massing's Galerkin course notes](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_02/galerkin_methods.html).

1. What matrix, nonlinear residual or semidiscrete evolution system follows from the chosen bases? definition
2. Which quadrature and geometry approximations alter the exact tested equations? measurement

Implementation defects

Record fluxes, penalties, stabilisation terms and solver tolerances with their intended mathematical roles.

1. Which interface, penalty or stabilisation terms are present, and what consistency argument supports them? provenance
2. How small must the algebraic or nonlinear solve error be relative to the intended discretisation accuracy? action

Orthogonality and quasioptimality

Consistent conforming linear Galerkin gives a(u-u_h,v_h)=0; Céa's estimate additionally requires the stated boundedness and coercivity assumptions. [Massing's Galerkin course notes](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_02/galerkin_methods.html).

1. Does exact Galerkin orthogonality hold, or what consistency defect replaces it? measurement
2. Which coercivity, inf-sup or other stability result applies, and are its constants uniform under refinement and parameter changes? provenance
3. In which norm and under which solution regularity assumptions is an error bound justified? boundary

Error evidence and response

Record benchmark errors, estimators, observed refinement behaviour and the specific decision each supports.

1. What manufactured, exact or independently resolved solution supports the reported convergence behaviour? provenance
2. Which measured norm or quantity-of-interest error meets the application's acceptance threshold? measurement
3. Does the evidence call for refinement, better integration, a tighter solve or a change of spaces or formulation? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of Galerkin method this model covers, and on what evidence? provenance

Kinds and varieties

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these kinds and varieties hold for the sense of Galerkin method this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of Galerkin method this model covers, and on what evidence? provenance

Typical measurements

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these typical measurements hold for the sense of Galerkin method this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of Galerkin method this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of Galerkin method this model covers, and on what evidence? provenance