Convergence estimates for multigrid algorithms with SSC smoothers and applications to overlapping domain decomposition

Aulisa, Eugenio; Bornia, Giorgio; Calandrini, Sara; Capodaglio, Giacomo

doi:10.1016/j.apnum.2018.04.011

Cited by 4 publications

(4 citation statements)

References 41 publications

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“…By starting with an initial guess u 0 ∈ H 1 0 (Ω) after a smoothing step which consists of a sequence of m local solutions over Ω i for i = 1, … , m we have from (18) that…”

Section: Proofmentioning

confidence: 99%

“…This alternative refinement strategy avoids the difficulties associated with hanging nodes but it leads to a larger support as compared to the conventional approach. Another very important group of domain decomposition solvers is the class of overlapping methods [15][16][17][18]. For instance, the class of Schwarz alternating methods is very reliable, robust and easy to implement.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A multigrid local smoother approach for a domain decomposition solver over non‐matching grids

Bornia

Chierici

Chirco

et al. 2021

Numerical Methods Partial

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In this paper we consider a multigrid approach for solving elliptic equations over non‐matching grids with domain decomposition methods. The domain is partitioned into subdomains with different mesh levels that do not match at the interface. The proposed algorithm searches for the global solution over different levels by projecting the residuals on the overlap region. This method is used in conjunction with a domain decomposition solver which only requires, in each iteration step, the solutions of several small local subproblems over finite element blocks. This algorithm is shown to converge to the solution of the corresponding Lagrange multiplier problem for non‐matching grids. The convergence properties of the algorithms are analyzed and numerical examples are presented. When the multigrid and domain decomposition approaches are combined, the method is shown to be reliable and easy to implement. Furthermore the local nature of the solver allows for a straightforward implementation on multiple parallel computers and graphics processing unit (GPU) clusters.

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“…By starting with an initial guess u 0 ∈ H 1 0 (Ω) after a smoothing step which consists of a sequence of m local solutions over Ω i for i = 1, … , m we have from (18) that…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A multigrid local smoother approach for a domain decomposition solver over non‐matching grids

Bornia

Chierici

Chirco

et al. 2021

Numerical Methods Partial

Self Cite

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“…In fact, by Definition 2.8, on E l k the function ϕ k,i is either zero or equal the function ϕ l k,i which is continuous and satisfies the delta property (2). We now have to introduce a more complex idea than the concept of master and hanging nodes.…”

Section: Construction Of the Uniform Finite Element Spacesmentioning

confidence: 99%

“…With this choice, special nodes called hanging nodes are introduced [14] and the finite element solution ceases to be continuous, unless extra care is exerted to prevent this. For instance, continuity could be enforced using a multilevel approach [40,2]. More frequently, modified basis functions for the finite element spaces are considered.…”

mentioning

confidence: 99%

Construction of H-Refined Continuous Finite Element Spaces with Arbitrary Hanging Node Configurations and Applications to Multigrid Algorithms

Aulisa¹,

Capodaglio²,

Ke³

2019

SIAM J. Sci. Comput.

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We present a novel approach for the construction of basis functions to be employed in selective or adaptive h-refined finite element applications with arbitrary-level hanging node configurations. Our analysis is not restricted to 1-irregular meshes, as it is usually done in the literature, allowing our results to be applicable to a broader class of local refinement strategies. The proposed method does not require the solution of any linear system to obtain the constraints necessary to enforce continuity of the basis functions and it can be easily implemented. A mathematical analysis is carried out to prove that the proposed basis functions are continuous and linearly independent. Finite element spaces are then defined as the spanning sets of such functions, and the implementation of a multigrid algorithm built on these spaces is discussed. A spectral analysis of the multigrid algorithm highlights superior convergence properties of the proposed method over existing strategies based on a local smoothing procedure. Finally, linear and nonlinear numerical examples are tested to show the robustness and versatility of the multigrid algorithm.restrict the residual to V J r i J = R J r i J ; 6. Let D J be an "easy to invert" approximation of the matrix A J , then solve D J w i J = r i J , and set u i+1 J = u i J + w i J ; 7. if w i J ≤ ε exit, otherwise set i = i + 1 and go back to step 3. A few remarks on the above algorithm:We construct the matrix A J and the residual r J in V J , since the assembling is done element-wise following the standard finite element approach for unstructured grids. Moreover, let x i , x j in X l J be two nodal points belonging to a generic element E of the triangulation T l J . The two pairs of test functions ϕ J,i , ϕ J,j and ϕ l J,i , ϕ l J,j EUGENIO AULISA, GIACOMO CAPODAGLIO, GUOYI KE PETSc [4] as a black box, providing as inputs the matrices A k for k = J, . . . , 0, the matrices Q k for k = J, . . . , 1 and the residual vector r i J , and letting PETSc compute w i J . This algebraic approach is convenient and elegant since all the burden of dealing with the hanging nodes at different level triangulations T k has been incorporated in the prolongation operator Q k and no ad-hoc multigrid solver or preconditioner needs to be implemented.The above algorithm can be easily extended to nonlinear problems. In this case the residual r i J at step 4 is a more complex function of u i J and D J is an "easy to invert" approximation of the tangent matrixThis corresponds to a Newton-Raphson iterative scheme [24]. Such a scheme is used in the last example where a Navier-Stokes flow is considered.

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An improved multigrid algorithm for n-irregular meshes with subspace correction smoother

Aulisa

Calandrini

Capodaglio

2018

Computers & Mathematics with Applications

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Convergence estimates for multigrid algorithms with SSC smoothers and applications to overlapping domain decomposition

Cited by 4 publications

References 41 publications

A multigrid local smoother approach for a domain decomposition solver over non‐matching grids

A multigrid local smoother approach for a domain decomposition solver over non‐matching grids

Construction of H-Refined Continuous Finite Element Spaces with Arbitrary Hanging Node Configurations and Applications to Multigrid Algorithms

An improved multigrid algorithm for n-irregular meshes with subspace correction smoother

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