Negative norm least-squares for div-curl systems.
by James H. Bramble, Joseph E. Pasciak, and Xuejun Zhang

Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
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In this talk, I will describe a least-squares approach for solving div-curl systems. Div-curl systems arise, for example, in electromagnetic applications. These systems are troublesome as they are not elliptic and a crude counting indicates that there are more equations than unknowns. By introducing appropriate negative norms, we provide a re-norming of $(L^2(\Omega))^d$ where $d=2$ or $d=3$ is the spatial dimension. This renorming is the basis for the least-squares method. The critical component of the method requires replacement of the negative norm by a computable discretization. Utilizing a vector decomposition given by Pasciak and Zhao, we are able to develop a computable discretization of the negative norm which only requires a preconditioner for second order problems and some simple vector operations.