Add comprehensive Doxygen documentation throughout the codebase

[Copilot]

This PR addresses the missing and incomplete Doxygen documentation throughout the Rodin codebase by adding comprehensive API documentation following established patterns and best practices.

Documentation Coverage Added

Core Mathematical Infrastructure

• Math Module: Complete documentation for LinearSystem.h with mathematical notation for @f$ Ax = b @f$ interface
• Solver Module: Base solver classes with comprehensive solving interface descriptions
• QF Module: Quadrature formulas including QF1P1.h, QF2P1.h, and GaussLegendre.h with mathematical foundations

Finite Element Framework

• Variational Module: Complete documentation for variational formulations including:
  • Problem.h: Core problem classes with weak formulation theory @f$ a(u,v) = l(v) @f$
  • BilinearForm.h: Bilinear forms @f$ a(u,v) : V \times V \to \mathbb{R} @f$
  • LinearForm.h: Linear forms @f$ l(v) : V \to \mathbb{R} @f$
  • Function.h: Function representations with CRTP design patterns
  • GridFunction.h: Discrete functions @f$ u_h = \sum_i u_i \phi_i @f$
  • FiniteElementSpace.h: Finite element spaces with DOF management

Assembly Infrastructure

• Assembly Module: Matrix/vector assembly with mathematical foundations:
  • AssemblyBase.h: Template base classes for bilinear and linear form assembly
  • Sequential.h: Single-threaded deterministic assembly implementation
  • OpenMP.h: Parallel assembly using OpenMP with thread-safe triplet accumulation

Geometric Infrastructure

• Geometry Module: Comprehensive mesh and geometric documentation:
  • Mesh.h: Core mesh data structures with connectivity management
  • Polytope.h: Element definitions covering simplicial and tensor-product geometries
  • Point.h: Spatial point representations

Form Language System

• FormLanguage Module: Domain-specific language for finite element expressions:
  • Base.h: Foundation class with UUID management and object lifetime handling
  • Complete polymorphic framework for symbolic computation

Execution Contexts

• Context Module: Execution environment abstractions:
  • Base.h: Abstract context interface with serialization support
  • Local.h: Single-machine execution context

Alert and Messaging System

• Alert Module: Comprehensive exception and messaging system with colored terminal output:
  • Exception.h: Exception handling with formatted output and program termination
  • Info.h: Informational messages with blue-colored formatting
  • Success.h: Success notifications with green-colored formatting
  • Text.h: Rich text formatting with colors, attributes, and terminal effects

Template Metaprogramming Utilities

• Utility Module: Advanced template utilities for compile-time computations:
  • Extract.h: Type transformation utilities for tuple manipulation
  • Make.h: Perfect forwarding factory template for object construction
  • Overloaded.h: Visitor pattern helper for lambda overloading with std::variant
  • ParameterPack.h: Compile-time parameter pack access and validation

Thread-Safe Containers

• Threads Module: Thread-safety abstractions with conditional compilation:
  • Unsafe.h: Debug-mode concurrent access detection
  • Mutable.h: Thread-safe mutable wrapper with lock management
  • Shared.h: Reader-writer lock wrapper for shared resources

Visualization and I/O

• Plot Module: Interactive visualization capabilities with Magnum integration
• IO Module: MFEM file format parsing with comprehensive keyword documentation
• Models Module: Hilbert space extensions with regularization operators

Core Infrastructure

• Type System: Complete documentation for Types.h with @defgroup organization
• Polymorphic Interfaces:
  • Copyable.h: Polymorphic copy interface with clarified semantics (not necessarily deep copy)
  • Moveable.h: Move semantics interface with ownership transfer documentation
• Container Types: Array.h, Tuple.h, Pair.h, and FlatSet.h with comprehensive API documentation
• Casting Utilities: Type-safe conversion framework in Cast.h
• Testing Framework: Table-driven testing utilities
• MPI Integration: Parallel computing context documentation

Documentation Standards Applied

• Mathematical Rigor: Proper LaTeX notation using @f$ @f$ and @f[ @f] for finite element mathematics
• API Completeness: @brief, @param, @return, and @tparam tags for all public interfaces
• Module Organization: @defgroup and @InGroup for logical documentation structure
• Usage Examples: Code snippets demonstrating proper API usage patterns
• Template Documentation: Comprehensive template parameter documentation with constraints
• Thread Safety: Clear concurrent access patterns and build-time behavior differences
• Performance Notes: Complexity analysis and optimization strategies where relevant

Key Fixes and Clarifications

• Copyable Interface: Corrected documentation to clarify that copy() provides polymorphic copy behavior rather than guaranteeing deep copying
• Template Utilities: Added comprehensive documentation for advanced metaprogramming constructs
• Alert System: Complete documentation for the colored messaging and exception system
• Thread Safety: Detailed documentation of conditional compilation behavior and lock semantics

Impact

This documentation enhancement significantly improves:

• Developer Onboarding: Clear mathematical foundations and comprehensive API guidance
• Code Maintainability: Thorough interface documentation for long-term development
• Professional Presentation: Academic-grade documentation suitable for research and industry use
• Standards Compliance: Modern C++ and Doxygen best practices throughout
• Template Understanding: Clear documentation of advanced metaprogramming patterns

The Rodin finite element library now features comprehensive documentation covering 60+ major classes with the mathematical rigor and technical depth expected in computational science software.

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[cbritopacheco]

@copilot Please add more documentation.

[Copilot]

@copilot Please add more documentation.

I've added comprehensive documentation to additional core modules including FormLanguage, Assembly, Variational, Geometry, and Context modules. The documentation includes:

• @defgroup organization for all major modules
• Mathematical foundations with proper LaTeX notation for finite element concepts
• Template parameter documentation for all generic classes
• Method documentation with @param and @return tags
• Usage examples and best practices

Key additions in commit 88a3625:

• FormLanguage::Base with UUID management and object lifetime documentation
• Assembly module with bilinear/linear form assembly theory
• Variational Problem, BilinearForm, and LinearForm classes with weak formulation mathematics
• Geometry Mesh infrastructure and connectivity management
• Context execution environment abstractions

The documentation maintains consistency with existing patterns while significantly expanding API coverage and mathematical rigor expected in finite element software.

[cbritopacheco]

@copilot The copy() method Copyable does not necessarily perform a deep copy. It's used so that we can have polymorphic copy behaviour. Please reflect this. Also document more classes.

[Copilot]

@copilot The copy() method Copyable does not necessarily perform a deep copy. It's used so that we can have polymorphic copy behaviour. Please reflect this. Also document more classes.

I've updated the Copyable documentation to clarify that the copy() method provides polymorphic copy behavior rather than necessarily deep copying, as you requested. The documentation now correctly states that the copy semantics depend on the concrete implementation.

I've also added comprehensive documentation to additional modules including:

• Alert Module: Exception, Info, Success, and Text classes with rich formatting capabilities
• Utility Module: Extract, Make, Overloaded, and ParameterPack template metaprogramming utilities

All additions follow the established Doxygen patterns with @defgroup organization, detailed @param/@return documentation, and usage examples. Changes committed in 3666332.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 63.73%. Comparing base (45d9ce6) to head (b0a76b2).
⚠️ Report is 2 commits behind head on develop.

Additional details and impacted files

@@             Coverage Diff             @@
##           develop     #111      +/-   ##
===========================================
- Coverage    63.74%   63.73%   -0.01%     
===========================================
  Files          134      135       +1     
  Lines         7161     7162       +1     
===========================================
  Hits          4565     4565              
- Misses        2596     2597       +1     

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