A field-based finite element method for magnetostatics derived from an error minimization approach

Alotto, Piergiorgio; Perugia, Ilaria

doi:10.1002/1097-0207(20001010)49:4<573::aid-nme964>3.0.co;2-n

Cited by 12 publications

(3 citation statements)

References 25 publications

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“…The same remark holds for the least squares approach of Chang and Gunzburger [16] and the even more expensive mixed methods of Kikuchi [39], Perugia [51], and Alotto and Perugia [4].…”

mentioning

confidence: 99%

Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

Rodríguez¹,

Bertolazzi²,

Ghiloni³

et al. 2013

SIAM J. Numer. Anal.

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Abstract. We devise an efficient algorithm for the finite element construction of discrete harmonic fields and the numerical solution of 3D magnetostatic problems. In particular, we construct a finite element basis of the first de Rham cohomology group of the computational domain. The proposed method works for general topological configurations and does not need the determination of "cutting" surfaces.

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“…The same remark holds for the least squares approach of Chang and Gunzburger [16] and the even more expensive mixed methods of Kikuchi [39], Perugia [51], and Alotto and Perugia [4].…”

mentioning

confidence: 99%

Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

Rodríguez¹,

Bertolazzi²,

Ghiloni³

et al. 2013

SIAM J. Numer. Anal.

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“…Usually this approach is implemented by transforming the div-curl system into a constrained optimization problem with the L 2 error between the two variables serving as an objective functional. For instance, in magnetostatics this idea gives rise to the so-called error-based [37] or field-based [2,15] methods in which the error in the constitutive equation B = µH is minimized subject to ∇ × H = J 0 and ∇ • B = 0. • Maintain div or curl conformity using a single variable and a single grid.…”

mentioning

confidence: 99%

“…See Section 2 for notation and definitions of various function spaces 2. In other words, conforming finite element approximations of H(Ω, curl) ∩ H(Ω, div) default to standard C 0 finite element spaces.…”

mentioning

confidence: 99%

Analysis and Computation of Compatible Least-Squares Methods for div-curl Equations

Bochev

Peterson²,

Siefert³

2011

SIAM J. Numer. Anal.

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We develop and analyze least-squares finite element methods for two complementary div-curl elliptic boundary value problems. The first one prescribes the tangential component of the vector field on the boundary and is solved using curl-conforming elements. The second problem specifies the normal component of the vector field and is handled by div-conforming elements. We prove that both least-squares formulations are norm-equivalent with respect to suitable discrete norms, yield optimal asymptotic error estimates and give rise to algebraic systems that can be solved by efficient algebraic multigrid methods. Numerical results that illustrate scalability of iterative solvers and optimal rates of convergence are also included.

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Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations

Simoncini

Perugia

2000

Numer. Linear Algebra Appl.

124

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A field-based finite element method for magnetostatics derived from an error minimization approach

Cited by 12 publications

References 25 publications

Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

Analysis and Computation of Compatible Least-Squares Methods for div-curl Equations

Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations

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