An Efficiency-Based Adaptive Refinement Scheme Appliedto Incompressible, Resistive MagnetohydrodynamicsJ. Adler1, T. Manteuffel1, S. McCormick1, J. Nolting1, J. Ruge1, and L. TangUniversity of Colorado at BoulderDepartment of Applied MathematicsBoulder, COAbstract. This paper describes the use of an efficiency-based adaptive mesh refinementscheme, known as ACE, on a 2D reduced model of the incompressible, resistive magnetohy-drodynamic (MHD) equations. A first-order system least squares (FOSLS) finite elementformulation and algebraic multigrid (AMG) are used in the context of nested iteration.The FOSLS a posteriori error estimates allow the nested iteration and ACE algorithms toyield the best accuracy-per-computational-cost. The ACE scheme chooses which elementsto add when interpolating to finer grids so that the best error reduction with the leastamount of cost is obtained, when solving on the refined grid. We show that these methods,applied to the simulation of a tokamak fusion reactor instability, yield approximations tosolutions within discretization accuracy using less than the equivalent amount of workneeded to perform 10 residual calculations on the finest uniform grid.Key words: magnetohydrodynamics, adaptive mesh refinement, algebraic multigrid, nestediteration1 IntroductionMagnetohydrodynamics (MHD) is a model of plasma physics that tr e ats the plasma as a chargedfluid. As a result, the set of partial differential eq uations that des cribe this model ar e a time-dependent, nonlinear system of equations. Thus, the equations can be difficult to solve andefficient numerical algor ithms ar e needed. This pa per shows the use of such an efficient algorithmon the incompressible, resistive MHD equations. A first-order systems leas t-squares (FOSLS)[1, 2] finite element discretization is used along with nested iteration and algebraic multigrid(AMG) [3–8]. The main focus of this paper is to show that if an efficiency-based adaptive meshrefinement (AMR) scheme is used, within the nested iteration algorithm, then a nonlinea r systemof equations, such as the MHD equations, can be solved in only a handful of work units pertime step. Here, a work unit is defined as the equivalent of one rela xation sweep on the finestgrid. In other words, the accuracy-per-co mputational-cost for solving the MHD equations can bemaximized by the use of nested iteration and AMR. As is shown in the results section, we wereable to resolve an island coalescence instability in less than 10 work units per time step.The MHD system and the FOSLS methodology applied to it are discussed in detail in thecompanion paper [9], so we include only a brief descr iption here in section 2. The nested iterationalgorithm has also been describe d in [10] and [11], so we only briefly discuss it here in section3. This section also discusses the efficiency-based AMR method known as ACE, which wasdeveloped in [1 2–14]. Finally, in section 4, numerical results are shown for a 2D reduced modelthat simulates plasma instabilities in a tokamak reactor. These results confirm that the AMRalgorithm greatly reduces the amount of work needed to solve the MHD systems.2 Adler, Manteuffel, McCormick, Nolting, Ruge, and Tang2 The MHD equations and FOSLS formulat ionThe resistive MHD equations are time-dependent and nonlinear, and involve several dependentvariables. The system is a coupling of the inco mpressible Navier-Stokes and Maxwell’s s ystems.The primitive variables are defined to be the fluid velocity, u, the fluid pressure, p, the magneticfield, B, the current density, j, and the electric field, E. In addition, a resistive form of Ohm’slaw,j = σ(E + u × B), (1)is used to eliminate the elec tric field, E, from the equations. After a non- dimensionalizationusing Alfv´en units, the following equations for incompressible resistive MHD are obtained (i.e.,Navier-Stokes coupled with Maxwell’s equations) [15, 16]:∂u∂t+ u ·∇u − j × B + ∇p −1Re∇2u = f , (2)∂B∂t− B · ∇u + u · ∇B +1SL(∇ × j) = g, (3)∇ × B = j, (4)∇ · B = 0, (5)∇ · u = 0, (6)∇ · j = 0. (7)Here, Reis the fluid Reynolds Number and SLis the Lundquist Number, both of which areassumed to be constants and adjusted for different types of physical behavior.Using the FOSLS method [1, 2], the system is first put into a differential first-order systemof equations. This is done based on a vorticity-velocity-pressure-current formulation [17–19]. Ascaling analysis is performed in [9] for the full MHD system. The resulting scaling yields a niceblock structure of the MHD system, which results in good AMG convergence of the linear systemsobtained, while still preserving the physics of the sy stem.Vorticity, ω = ∇ × u, is introduced and the final formulation in 3D used is1√Re∇ × u −pReω = 0, (8)1√Re∇ · u = 0, (9)pRe∇ · ω = 0, (10)1√Re∂u∂t− u × ω − j × B −pRe∇p +1√Re∇ × ω = f , (11)1√SL∇ × B −pSLj = 0, (1 2)1√SL∇ · B = 0, (13)pSL∇ · j = 0, (14)1√SL∂B∂t+1√ReSL(u · ∇B − B · ∇u) +1√SL∇ × j = g. (15)An AMR S cheme for Incompressible RMHD 3The above system is denoted by L(u) = f , where u = (u, ω, p, B, j)Trepresents a vector of allof the dependent variables that should not be confused with the vector fluid velocity, u. Then,the L2norm of the residual of this system is minimized. This is referred to as the nonlinearfunctional,F(u) = ||L(u) − f||0. (16)In ge neral, we wish to find the arg min of (16) in some so lution space V. In the context of thispaper, we choose V to be an H1product space with boundary conditions that are cho sen tosatisfy the phy sical constra ints of the problem as well as the assumptions needed for the FOSLSframework. In practice, a series of nested finite subspac e s, Vh, are used to approximate thesolution in V. However, in the Newton-FOSLS approach [20, 21], system (8)-(15) is first linearizedusing a Newton step before a FOSLS functional is formed and minimized. This results in theweak form of the pro blem that produces symmetric positive definite algebraic systems when theproblem is restricted to a finite-dimensional subspace, Vh. In addition, proving continuity andcoercivity of the resulting bilinear form is equivalent to having H1equivalence of the FOSLSfunctional. Mo reover, the FOSLS functional yields a sharp a posteriori local error estimate,which is used to make the algorithm more robust and, under the rig ht conditions,