Let's break down the solution into **2 main steps** for the finite element method (FEM) for the given Laplace equation. --- ## **Step 1: Discretization and Assembly of Element Matrices** ### **Explanation:** - The domain is discretized into **4 linear rectangular elements** as shown in the mesh. - Each node will have an associated degree of freedom (unknown nodal value of \( u \)), except where Dirichlet boundary conditions are applied. - The weak form of the equation is assembled for each element and then combined into a global system. #### **Element Stiffness Matrix (for a rectangular element):** For a linear rectangular element with side lengths \( h_x \) and \( h_y \): \[ K^{(e)} = \frac{1}{6} \begin{bmatrix} 2 & -1 & -1 & 0 \\ -1 & 2 & 0 & -1 \\ -1 & 0 & 2 & -1 \\ 0 & -1 & -1 & 2 \\ \end{bmatrix} \frac{h_y}{h_x} + \frac{1}{6} \begin{bmatrix} 2 & -1 & -1 & 0 \\ -1 & 2 & 0 & -1 \\ -1 & 0 & 2 & -1 \\ 0 & -1 & -1 & 2 \\ \end{bmatrix} \frac{h_x}{h_y} \] - **For \( a = b = 1 \) and \( h_x = h_y = 0.5 \)**, you substitute these values. - The element matrices are assembled into a **global stiffness matrix** corresponding to the numbering of the nodes. --- ## **Step 2: Application of Boundary Conditions & Solution of the System** ### **Explanation:** - **Apply Dirichlet boundary conditions:** For nodes on the boundaries where \( u \) is specified (e.g., \( u = 0 \) or \( u(x, b) = u_0(x) \)), set those unknowns directly and modify the system accordingly. - For nodes where Neumann or natural boundary conditions are specified (e.g., \( \frac{\partial u}{\partial x} = 0 \)), incorporate these into the global load vector if needed. - **Solve the resulting system:** The modified global stiffness matrix and load vector yield a linear system for the unknown nodal values. --- ## **Final Answer** **The approximate solution \( u(x, y) \) is obtained by:** 1. **Assembling the global stiffness matrix** using the given mesh and element matrices, based on the Laplace equation and the mesh of 4 linear rectangular elements. 2. **Applying the boundary conditions**, reducing the system to the unknowns at the interior nodes, and then **solving the resulting linear system** to obtain the nodal values of \( u \). **These nodal values provide the approximate solution to the Laplace equation over the domain according to the finite element discretization.** --- If you need explicit calculations (e.g., the global matrix or nodal values for a specific right-hand side), please specify the boundary function \( u_0(x) \) or any source term.

Certainly! Here's a detailed, step-by-step solution for approximating the solution to the Laplace equation using the finite element method (FEM) with 4 rectangular elements, based on the provided problem data. --- ## **Step 1: Discretization and Assembly of Element Matrices** ### **1.1 Domain and Mesh Details** - The domain is a rectangle with dimensions \( a \times b \). - The mesh consists of 4 linear rectangular elements arranged in a grid. - The nodes are numbered, and boundary conditions are applied at specified nodes. *Note:* Since the exact coordinates and numbering are not explicitly provided in the image, we assume a standard uniform mesh with nodes numbered from bottom-left to top-right. ### **1.2 Element Stiffness Matrix** For a rectangular element with sides \( h_x \) and \( h_y \), the local stiffness matrix \( K^{(e)} \) (assuming linear shape functions) is: \[ K^{(e)} = \frac{1}{h_x h_y} \begin{bmatrix} h_y + h_x & -h_x & -h_y & 0 \\ -h_x & h_x + h_y & 0 & -h_y \\ -h_y & 0 & h_y + h_x & -h_x \\ 0 & -h_y & -h_x & h_x + h_y \end{bmatrix} \] *Given:* \( a = b = 1 \), and mesh size \( h_x = h_y = 0.5 \). Calculating: \[ K^{(e)} = \frac{1}{0.5 \times 0.5} \times \begin{bmatrix} 0.5 + 0.5 & -0.5 & -0.5 & 0 \\ -0.5 & 1 & 0 & -0.5 \\ -0.5 & 0 & 1 & -0.5 \\ 0 & -0.5 & -0.5 & 0.5 + 0.5 \end{bmatrix} = 4 \times \begin{bmatrix} 1 & -0.5 & -0.5 & 0 \\ -0.5 & 1 & 0 & -0.5 \\ -0.5 & 0 & 1 & -0.5 \\ 0 & -0.5 & -0.5 & 1 \end{bmatrix} = \begin{bmatrix} 4 & -2 & -2 & 0 \\ -2 & 4 & 0 & -2 \\ -2 & 0 & 4 & -2 \\ 0 & -2 & -2 & 4 \end{bmatrix} \] ### **1.3 Assembly into Global Stiffness Matrix** - Map local nodes to global nodes. - Assemble local matrices into the global stiffness matrix \( K \). *Note:* Without explicit node numbering, the general process is: 1. For each element, insert \( K^{(e)} \) into the global matrix at positions corresponding to the element's nodes. 2. Sum contributions at shared nodes. --- ## **Step 2: Application of Boundary Conditions & Solving** ### **2.1 Boundary Conditions** - \( u = 0 \) on the boundary where specified (e.g., \( u=0 \) at boundary nodes). - Given boundary conditions: \[ u(0,y) = 0,\quad u(a,y) = 0,\quad u(x,0) = 0,\quad u(x,b) = u_0(x) = \sin(x + 0.5a) \] - Assign known \( u \) values at boundary nodes and modify the global system accordingly. ### **2.2 Modifying the System** - For nodes with prescribed \( u \), set the corresponding equations to enforce boundary values. - Adjust the load vector (if any source terms are present, which in Laplace's equation typically is zero). ### **2.3 Solving the Linear System** - Solve the resulting system \( K \mathbf{u} = \mathbf{f} \) for the unknown nodal values. - Use standard linear algebra techniques (Gaussian elimination, LU decomposition, etc.). --- ## **Final Result:** - The **approximate solution** \( u(x, y) \) is reconstructed from the nodal values obtained from solving the system. - The solution satisfies the Laplace equation approximately over the domain, with boundary conditions applied. --- **Summary:** 1. **Discretize** the domain into 4 rectangular elements, derive local stiffness matrices, and assemble into a global matrix. 2. **Apply boundary conditions** to modify the system. 3. **Solve** the linear system for nodal values. 4. **Interpolate** to find \( u(x,y) \) within each element. --- If specific node coordinates, boundary node indices, or source terms are provided, more explicit numerical solutions can be computed.