We approach the problem of solving large-scale, extremelly ill-conditioned banded linear systems using direct methods. Unlike traditional approaches, we focus on limiting the memory footprint of the algorithms rather than the FLOP count. To reduce the memory demand, BLAS-3 pre- and post-processing of the linear system are required. While this increases considerably the number of calculations required to solve the system, most of this work can be done very efficiently on the GPU. In this way, using GPUs allows us to solve much larger problems than state-of-the-art banded direct solvers on modern architectures. We'll present results for problems arising from realistic oil and gas scenarios, and we'll show that these techniques allow us to solve systems of tens of millions of equations using significantly less memory than currently available direct banded solvers.