DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

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Rectangular grid and Dirichlet flux stencil

Definition

Let $L_{x}, L_{y} > 0$ , let $N_{x}, N_{y}$ be integers at least $2$ , and set $h_{x} = L_{x} / N_{x}$ and $h_{y} = L_{y} / N_{y}$ . The rectangular grid nodes are $(x_{i}, y_{j}) = (i h_{x}, j h_{y})$ for $0 \leq i \leq N_{x}$ and $0 \leq j \leq N_{y}$ . The interior index set is $I_{h} = {(i, j) : 1 \leq i < N_{x}, 1 \leq j < N_{y}}$ , and its boundary is the set of grid nodes with $i \in {0, N_{x}}$ or $j \in {0, N_{y}}$ . A grid function is a real-valued function on all grid nodes; $X_{h, 0}$ is the vector space of grid functions that vanish on the boundary.

For real vectors $v, w$ indexed by $I_{h}$ , define the weighted interior inner product and norm by

⟨ v, w ⟩_{h} = h_{x} h_{y} \sum_{i = 1}^{N_{x} - 1} \sum_{j = 1}^{N_{y} - 1} v_{i j} w_{i j}, ∥ v ∥_{h} = ⟨ v, v ⟩_{h}^{1 / 2} .

For full grid functions, the same notation applies to their restrictions to $I_{h}$ ; it gives a semidefinite bilinear form and a seminorm on all grid functions, and an inner product and norm on $X_{h, 0}$ .

The edge differences are $D_{x} v_{i + 1 / 2, j} = (v_{i + 1, j} - v_{i j}) / h_{x}$ for $0 \leq i < N_{x}$ , $1 \leq j < N_{y}$ , and $D_{y} v_{i, j + 1 / 2} = (v_{i, j + 1} - v_{i j}) / h_{y}$ for $1 \leq i < N_{x}$ , $0 \leq j < N_{y}$ . For a scalar coefficient $a$ sampled at face midpoints, write $a_{i + 1 / 2, j} = a (x_{i} + h_{x} / 2, y_{j})$ and $a_{i, j + 1 / 2} = a (x_{i}, y_{j} + h_{y} / 2)$ .

The divergence-form Dirichlet operator on interior nodes is

(A_{h}^{a} v)_{i j} = - \frac{a_{i + 1 / 2, j} D_{x} v_{i + 1 / 2, j} - a_{i - 1 / 2, j} D_{x} v_{i - 1 / 2, j}}{h_{x}} - \frac{a_{i, j + 1 / 2} D_{y} v_{i, j + 1 / 2} - a_{i, j - 1 / 2} D_{y} v_{i, j - 1 / 2}}{h_{y}} .

For $a \equiv 1$ , this is the five-point Dirichlet Laplacian $- D_{x}^{+} D_{x}^{-} - D_{y}^{+} D_{y}^{-}$ . The associated edge energy is

E_{h}^{a} (v, w) = h_{x} h_{y} (\sum_{j = 1}^{N_{y} - 1} \sum_{i = 0}^{N_{x} - 1} a_{i + 1 / 2, j} D_{x} v_{i + 1 / 2, j} D_{x} w_{i + 1 / 2, j} + \sum_{i = 1}^{N_{x} - 1} \sum_{j = 0}^{N_{y} - 1} a_{i, j + 1 / 2} D_{y} v_{i, j + 1 / 2} D_{y} w_{i, j + 1 / 2}) .

The homogeneous scheme solves $A_{h}^{a} v = f_{h}$ in $I_{h}$ with $v = 0$ on the boundary. For Dirichlet data $g$ , it solves the same interior equations with $v = g$ at boundary nodes. This definition specifies the rectangular Dirichlet stencil; it does not specify a Neumann or nonrectangular boundary closure.

Used by

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Sources

Endre Süli, Numerical Solution of PDEs, complete lecture notes (standard reference, not scraped)
Benjamin Seibold, MIT 18.336 Lecture 5: Finite Difference Approximation (standard reference, not scraped)
Benjamin Seibold, MIT 18.336 Lecture 6: General Linear Second Order Equation (standard reference, not scraped)