Mixed  and  least‐squares finite element methods with application to linear hyperbolic problems

Kalchev, Delyan Z.; Manteuffel, Thomas A.; Münzenmaier, Steffen

doi:10.1002/nla.2150

Cited by 6 publications

(12 citation statements)

References 40 publications

(81 reference statements)

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“…In contrast, Theorem 5.11 suggests that handling the globally worst case is not necessary, since, in view of (5.5), formulation (4.11) enforces certain “proximity” of

ψ_{u^{h}}

to the discrete space

{scriptV}_{Γ_{I}}^{h}

, enhancing the control of the L 2 (Ω) norm. Intuitively, this means that

{scriptV}_{Γ_{I}}^{h}

“filters out” modes with high‐frequency cross stream oscillations, like the ones in [28], section 3.7.1, that can cause the lack of uniform L 2 coercivity. That is, the proximity of

ψ_{u^{h}}

{scriptV}_{Γ_{I}}^{h}

, measured by

R_{u^{h}}

, enhances the control of the

H_{0, Γ_{I}}^{1} (Ω)

component of the Helmholtz decomposition (4.5), while the

H_{0, Γ_{C}}^{1} (Ω)

component is naturally controlled by

F

.…”

Section: Discussionmentioning

confidence: 99%

“…This is associated with the relation to (3.3) and (4.2); compare, (4.4) and Remark 4.4. Moreover, even in the linear case, one can construct an oscillatory counterexample (see [28]) to demonstrate the lack of an appropriate uniform coercivity.…”

Section: Discussionmentioning

confidence: 99%

“…This paper focuses on least‐squares finite element techniques. Least‐squares methods [14] have been developed for a variety of problems, including linear [7, 15–17] and nonlinear [18] first‐order hyperbolic PDEs; see also [19–21]. Those approaches utilize appropriate least‐squares minimization principles to obtain finite element discretizations of PDEs.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, a “restricted” version of the inf‐sup condition is suggested, that improves on the convergence result (Theorem 5.11). The implicit mesh‐dependent L 2 coercivity was discussed in [28]. Also, a spacial functional regularization (Section 5.2) is proposed that explicitly provides (mesh‐dependent) L 2 coercivity (Theorem 5.7).…”

Section: Introductionmentioning

confidence: 99%

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A least‐squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

Kalchev

Manteuffel

2020

Numerical Methods Partial

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In this paper, a least-squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative-norm least-squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax-Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L 2 convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least-squares functional is continuous and coercive in an H −1-type norm, but not L 2-coercive. Nevertheless, the L 2 convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L 2 norm, or by properly choosing the finite element spaces, providing implicit control of the L 2 norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L 2 convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least-squares functional, minimized on finite element spaces, and a Gauss-Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this

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“…In contrast, Theorem 5.11 suggests that handling the globally worst case is not necessary, since, in view of (5.5), formulation (4.11) enforces certain “proximity” of

ψ_{u^{h}}

to the discrete space

{scriptV}_{Γ_{I}}^{h}

, enhancing the control of the L 2 (Ω) norm. Intuitively, this means that

{scriptV}_{Γ_{I}}^{h}

“filters out” modes with high‐frequency cross stream oscillations, like the ones in [28], section 3.7.1, that can cause the lack of uniform L 2 coercivity. That is, the proximity of

ψ_{u^{h}}

{scriptV}_{Γ_{I}}^{h}

, measured by

R_{u^{h}}

, enhances the control of the

H_{0, Γ_{I}}^{1} (Ω)

component of the Helmholtz decomposition (4.5), while the

H_{0, Γ_{C}}^{1} (Ω)

component is naturally controlled by

F

.…”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A least‐squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

Kalchev

Manteuffel

2020

Numerical Methods Partial

Self Cite

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show abstract

“…Kalchev et al study a few dual least‐squares finite element methods and their application to scalar linear hyperbolic problems. The purpose is to obtain L 2 ‐norm approximations on finite element spaces of the exact solutions to hyperbolic partial differential equations of interest.…”

mentioning

confidence: 99%