Even though these libraries are designed for sparse matrices, surely the space utilization of such a matrix would be crazy large in practice?
It seems like this matrix is not only sparse, but has a very regular/particular structure. Is there an even better way to solve the equation than using a generic sparse linear solver?
How exactly does accounting for the boundary in a special manner fix our linear system?
What is the complexity of this?
It seems that this would be very costly in practice. Is there another alternative to store the sparse matrix information that is more efficient in terms of storage?
Are there substantial advantages in complexity when solving these problems on sparse matrices? My intuition is that some of the operations on sparse matrices (like finding eigenvalues) might not be easier than on dense matrices, but things like solving linear equations, for example, might be.