Stable Time Stepping for Elliptic Problems via Lumped Mass Densities

In this article, we present a new method for solving partial differential equations (PDEs) in acoustic and electromagnetic simulations. Traditional methods require high-degree polynomial basis functions to achieve accurate solutions, but these can be computationally expensive. Our approach, called mass lumping, simplifies the PDEs by grouping similar terms together and treating them as a single entity. This reduces the number of computations required, making simulations faster and more efficient.
Imagine you’re trying to build a house with a complex roof design. Traditional methods would require you to create each individual piece of the roof separately, which can be time-consuming and expensive. Mass lumping is like creating a blueprint that groups all the pieces of the roof together into a single entity, making it easier and faster to build.
Our method works by dividing the PDEs into smaller, more manageable parts called "mass lumps." Each mass lump represents a group of terms in the PDEs that have similar properties, such as the same frequency or wave number. By treating these groups together, we can significantly reduce the computational cost without compromising accuracy.
We demonstrate the efficiency and accuracy of our method through numerical experiments using the finite element library Netgen/NGSolve [10, 11]. Our results show that mass lumping can achieve high-degree polynomial accuracy while reducing the number of computations required by up to two orders of magnitude. This makes it possible to solve PDEs in complex simulations, such as acoustic and electromagnetic wave propagation, much faster than before.
In summary, mass lumping is a powerful tool for solving PDEs that simplifies the computation process while maintaining accuracy. By grouping similar terms together, we can reduce the computational cost and make simulations faster and more efficient. This breakthrough approach has the potential to revolutionize the field of numerical analysis and scientific computing, enabling researchers to tackle complex problems in physics and engineering with unprecedented speed and accuracy.