Extension of Algorithm 2 to General d-Dimensional Case

In this article, we delve into the fascinating world of numerical solution of partial differential equations (PDEs). The journey begins with an introduction to the basics of PDEs and their importance in modeling various phenomena in physics, engineering, and other fields. We then shift our focus to the art of discretization, which is the process of approximating PDEs using numerical methods.
Discretization is like a game of chess, where we break down complex problems into smaller, manageable pieces called finite elements or volumes. Each element or volume represents a small part of the problem, and we use various techniques to approximate the solution within each one. The key to success lies in choosing the right discretization scheme, which is like selecting the right opening move in chess.
One popular discretization method is the finite difference method, which is like counting the number of soldiers on each square of a chessboard. By breaking down the problem into small enough squares, we can approximate the solution using simple arithmetic operations. Another approach is the finite element method, which involves dividing the problem domain into smaller elements and approximating the solution using basis functions. This method is like building a castle with Lego bricks, where each brick represents a small part of the structure.
The art of discretization also involves selecting the right numerical solver, which is like choosing the right weapon in a battle. Different solvers have their strengths and weaknesses, and we must select the one that best fits our problem at hand. Some solvers are more accurate but slower, while others are faster but less precise. Finding the balance between accuracy and speed is crucial for achieving successful solutions.
In addition to discretization techniques, we also explore various error estimation methods, which are like checking the weather forecast before embarking on a journey. By estimating errors, we can determine how accurate our solutions are and make necessary adjustments to improve their accuracy.
Throughout the article, we use everyday analogies and metaphors to demystify complex concepts and make them more accessible to readers. We also provide practical examples of numerical solution techniques for various PDEs, including the heat equation, wave equation, and Laplace equation. By the end of the article, readers will have a comprehensive understanding of the art of discretization and how it is used to solve real-world problems in science and engineering.