Multilevel First-Order System Least Squares for Nonlinear Elliptic Partial Differential Equations

Codd, A. L.; Manteuffel, Thomas A.; McCormick, Steve

doi:10.1137/s0036142902404406

Cited by 34 publications

(34 citation statements)

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“…A second aspect is the interstage scale factor ε. In [10], we discussed the two-stage algorithm, where, in the first stage, we set ε = 0 and solve for J n+1 and, in the second, we set ε = 1 and solve for x n+1 . The second stage amounts to a simple system of decoupled Poisson equations.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The method we use to obtain an approximation to D * (or J * ) is discussed in some detail in [10]. Here we give a brief overview.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…We are interested in the functional measure because it is equivalent to the H norm of the errors, as we established theoretically in [10] and as these graphs suggest. The left-hand graph contains the functional values, and the right-hand graph contains the errors.…”

Section: First-stage and Single-stage Algorithmsmentioning

confidence: 99%

“…For the test marked "unscaled" and for which h = Choices of the numbers of V(1,1)-cycles per iteration and Newton steps on each grid are currently made by observation. In the theoretical section of [10], we suggested ρ ν0 ≤ 1 8 as a criterion, where ρ is the convergence factor and ν 0 is the number of Vcycles. A significantly larger value would allow the iterates to wander too far from the true solution as the grid was refined.…”

Section: Amg Testsmentioning

confidence: 99%

“…The assumptions needed to apply the theory in [10] are verified in section 3. Section 4 discusses scaling of the functional terms used for the computations as well as numerical results for two representative problems.…”

mentioning

confidence: 99%

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Multilevel First-Order System Least Squares for Elliptic Grid Generation

Codd

Manteuffel²,

McCormick³

et al. 2003

SIAM J. Numer. Anal.

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Abstract.A new fully variational approach is studied for elliptic grid generation (EGG). It is based on a general algorithm developed in a companion paper [A. L. Codd, T. A. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 41 (2003), pp. 2197-2209 that involves using Newton's method to linearize an appropriate equivalent first-order system, first-order system least squares (FOSLS) to formulate and discretize the Newton step, and algebraic multigrid (AMG) to solve the resulting matrix equation. The approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse-level computation. The present paper verifies the assumptions of the companion work and confirms the overall efficiency of the scheme with numerical experiments.Key words. least-squares discretization, multigrid, nonlinear elliptic boundary value problems AMS subject classifications. 35J65, 65N15, 65N30, 65N50, 65F10 DOI. 10.1137/S00361429024044181. Introduction. A companion paper [10] develops an algorithm using Newton's method, first-order system least squares (FOSLS), and algebraic multigrid (AMG) for efficient solution of general nonlinear elliptic equations. The equations are first converted to an appropriate first-order system, and an approximate solution to the coarsest-grid problem is then computed (by any suitable method such as Newton iteration coupled perhaps with direct solvers, damping, or continuation). The approximation is then interpolated to the next finer level, where it is used as an initial guess for one Newton linearization of the nonlinear problem, with a few AMG cycles applied to the resulting matrix equation. This algorithm repeats itself until the finest grid is processed, again by one Newton/AMG step. At each Newton step, FOSLS is applied to the linearized system, and the resulting matrix equation is solved using just a few V-cycles of AMG.In the present paper, we apply this algorithm to elliptic grid generation (EGG) equations. Grid generation is usually based on a map between a relatively simple computational region and a possibly complicated physical region. It can be used numerically to create a mesh for a discretization method to solve a given system of equations posed on the physical domain. Alternatively, it can be used to transform equations posed on the physical region into ones posed on the computational region, where the transformed equations are then solved. If the Jacobian of the transformation is positive throughout the computational region, the equation type is unchanged [12]. Actually, the relative minimum value of the Jacobian is important in practice because relatively small values signal small angles between the grid lines and large errors in approximating the equations [20].

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Section: Numerical Resultsmentioning

confidence: 99%

“…The method we use to obtain an approximation to D * (or J * ) is discussed in some detail in [10]. Here we give a brief overview.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: First-stage and Single-stage Algorithmsmentioning

confidence: 99%

Section: Amg Testsmentioning

confidence: 99%

mentioning

confidence: 99%

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Multilevel First-Order System Least Squares for Elliptic Grid Generation

Codd

Manteuffel²,

McCormick³

et al. 2003

SIAM J. Numer. Anal.

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A first‐order system least‐squares finite element method for the Poisson‐Boltzmann equation

et al. 2009

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The Poisson-Boltzmann equation is an important tool in modeling solvent in biomolecular systems. In this article, we focus on numerical approximations to the electrostatic potential expressed in the regularized linear Poisson-Boltzmann equation. We expose the flux directly through a first-order system form of the equation. Using this formulation, we propose a system that yields a tractable least-squares finite element formulation and establish theory to support this approach. The least-squares finite element approximation naturally provides an a posteriori error estimator and we present numerical evidence in support of the method. The computational results highlight optimality in the case of adaptive mesh refinement for a variety of molecular configurations. In particular, we show promising performance for the Born ion, Fasciculin 1, methanol, and a dipole, which highlights robustness of our approach.

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Parallel adaptive mesh refinement for first‐order system least squares

Brezina

García

Manteuffel

et al. 2012

Numerical Linear Algebra App

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SUMMARY This paper develops new adaptive mesh refinement strategies for first‐order system least squares (FOSLS) in conjunction with algebraic multigrid (AMG) methods in the context of nested iteration (NI). The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements. To accomplish this, the refinement decisions at each refinement level are determined on the basis of minimizing the ‘accuracy‐per‐computational‐cost’ efficiency (ACE) measure that takes into account both error reduction and computational cost. The NI‐FOSLS‐AMG‐ACE approach produces a sequence of refinement levels in which the error is equally distributed across elements on a relatively coarse grid. Once the solution is numerically resolved, refinement becomes nearly uniform. Accommodations of the ACE approach to massively distributed memory architectures involve a geometric binning strategy to reduce communication cost. Load balancing begins at very coarse levels. Elements and nodes are redistributed using parallel quadtree structures and a space‐filling curve, which automatically ameliorates load balancing issues at finer levels. Numerical results show that the NI‐FOSLS‐AMG‐pACE approach is able to provide highly accurate approximations to rapidly varying solutions at relatively low cost. Excellent weak and strong scalability are demonstrated on 4096 processors for problems with 15 million biquadratic elements.Copyright © 2012 John Wiley & Sons, Ltd.

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Multilevel First-Order System Least Squares for Nonlinear Elliptic Partial Differential Equations

Cited by 34 publications

References 20 publications

Multilevel First-Order System Least Squares for Elliptic Grid Generation

Multilevel First-Order System Least Squares for Elliptic Grid Generation

A first‐order system least‐squares finite element method for the Poisson‐Boltzmann equation

Parallel adaptive mesh refinement for first‐order system least squares

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