5.3.2. Example: Solving a 1D Poisson Equation Using FEM

Divide the domain [0,L][0, L] into NN elements, each with nodes at the endpoints.

In this step, we take the continuous domain of the function we are trying to analyze, which is represented as an interval [0, L]. We split this interval into smaller, manageable pieces called 'elements'. Each of these elements has specific locations called 'nodes' at its endpoints. This process is essential for applying the Finite Element Method (FEM), as it allows us to consider the behavior of the solution at these discrete points rather than continuously across the entire interval.

Imagine you're trying to measure a long, straight road. Instead of measuring the whole road in one go, you decide to measure it in segments. You mark the start and end of each segment (the nodes) so that you can get precise values for each small part rather than just an estimate for the whole road.

For each element, choose linear shape functions N1(x) and N2(x) that interpolate the solution between the nodes.

After discretizing the domain, we need to approximate the solution within each of the elements. This is done by selecting linear shape functions, denoted as N1(x) and N2(x). These functions help us interpolate the solution value at any point within the element based on the values at the nodes. Essentially, they provide a way of representing the solution as a linear combination of the values at these nodes, allowing us to effectively model the behavior of the entire element.

Think about using connecting rods to create a bridge. The bridge might need to hold weight at various points (the nodes). The rods (shape functions) help distribute the weight evenly across the structure rather than concentrating it only at the joints. This way, we can understand how the whole bridge behaves based on the performance at specific support points.

Multiply the Poisson equation by a test function v(x) and integrate by parts.

In this step, we enhance our original Poisson equation by multiplying it by a function called a 'test function' v(x). This process modifies the equation, allowing us to convert it into a 'weak form'. Integrating by parts is a mathematical technique that helps reduce the order of derivatives in our equation, making it more suitable for numerical calculations. This weak form is crucial for the FEM as it aligns with the requirements of the approximation techniques used in solving PDEs.

Imagine you are distributing a load over a surface. Instead of applying the force directly at one point (which might create strain), you spread it out across a wider area using a flexible material (the test function). This way, you avoid stress concentrations, ensuring an even and controlled distribution of force – similar to how we distribute the mathematical properties in our equation.

The system of equations is constructed from the contributions of each element, leading to a global stiffness matrix and force vector.

Now that we have the weak form of the Poisson equation, we need to gather data from each individual element to form a complete system of equations. This is performed by summing up the contributions from each element, which results in a 'global stiffness matrix' and a 'force vector'. The stiffness matrix represents relations between the nodal values across the entire domain, while the force vector incorporates all the external factors acting on the system. This assembled system can then be solved to find the overall solution to the original problem.

Consider putting together a puzzle. Each piece represents an element, and when you put them all together (assemble), you create a complete picture (the global system). Each piece fits in a particular way to help show the overall scene, just like how contributions from each element build the entire solution to the Poisson equation.

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