By name: finite-element-discretization-integrals description: Calculate finite element discretization integrals (A.1, A.2, A.3) for converting governing equations into a system of ODEs. Use when performing spatial discretization using the finite element approach for drift-diffusion models or similar PDE systems requiring matrix assembly.
Finite Element Discretization Integrals
When to Use
Apply this skill when:
- Performing spatial discretization of governing equations using the finite element method
- Converting PDEs (specifically Eqs. 11-16) into a system of ordinary differential equations
- Assembling discretization matrices for numerical solution
- Working with drift-diffusion models that require integral evaluation over basis functions
Procedure
1. Define Basis Functions
- Use φ_i(x) as the basis "hat" or "tent" function (see Eq. 19 in references)
- Apply prime notation (') to denote derivatives with respect to x
- Set index range: i, j, k range from 0 to N
2. Calculate Required Integrals
Compute the following three integral types for the discretization matrix:
- Integral A.1: Required for primary matrix components
- Integral A.2: Required for secondary matrix components
- Integral A.3: Required for coupling terms and additional matrix elements
Refer to references/integral-forms.md for the specific mathematical expressions and evaluation methods.
3. Assemble Matrix Components
- Organize computed integrals into the system matrix structure
- Ensure proper index mapping for i, j, k
- Verify matrix properties match the governing equations
Key Variables
- φ_i(x): Basis "hat" function - piecewise linear interpolation function
- φ_i'(x): Derivative of basis function with respect to x
- i, j, k: Mesh indices (range: 0 to N)
- N: Number of mesh intervals
Output
Matrix components for the system of ODEs resulting from spatial discretization.