Existing algebraic multigrid (AMG) methods rely on assumptions about the near-kernel components of a given linear system. Namely, that these components are "smooth" in the sense that they can be sufficiently approximated by few degrees of freedom. PDEs with higher order terms violate these assumptions, causing an unbounded number of $V$-cycles for convergence. As an example, we introduce a PDE that arises in kinetic-edge plasma simulation. This PDE contains an isotropic fourth-order term, making existing methods infeasible. In this work, we propose an $O(n)$ highly-parallelizable exact method to solve the system solely containing the isotropic fourth-order term. We then extend this algorithm to solve the original system, including periodic boundary conditions. Our algorithm obtains drastic improvement over existing methods.
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