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• On a general implementation of h- and p-adaptive curl-conforming finite elements

  M. Olm, S. Badia, A. Martín

  Advances in Engineering Software (2019). Vol. 132, 74-91

  
  

  Abstract

  Edge (or Nédélec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, especially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR . We cover into detail how to implement arbitrary order (i.e., p-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., h-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation.

  Abstract
  Edge (or Nédélec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, especially [...]

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• Physics-based balancing domain decomposition by constraints for multi-material problems

  S. Badia, A. Martín, H. Nguyen

  Journal of Scientific Computing (2019). Vol. 79 (2), pp. 718-747

  
  

  Abstract

  In this work, we present a new variant of the balancing domain decomposition by constraints preconditioner that is robust for multi-material problems. We start with a well-balanced subdomain partition, and based on an aggregation of elements according to their physical coefficients, we end up with a finer physics-based (PB) subdomain partition. Next, we define corners, edges, and faces for this PB partition, and select some of them to enforce subdomain continuity (primal faces/edges/corners). When the physical coefficient in each PB subdomain is constant and the set of selected primal faces/edges/corners satisfy a mild condition on the existence of acceptable paths, we can show both theoretically and numerically that the condition number does not depend on the contrast of the coefficient across subdomains. An extensive set of numerical experiments for 2D and 3D for the Poisson and linear elasticity problems is provided to support our findings. In particular, we show robustness and weak scalability of the new preconditioner variant up to 8232 cores when applied to 3D multi-material problems with the contrast of the physical coefficient up to 108 and more than half a billion degrees of freedom. For the scalability analysis, we have exploited a highly scalable advanced inter-level overlapped implementation of the preconditioner that deals very efficiently with the coarse problem computation. The proposed preconditioner is compared against a state-of-the-art implementation of an adaptive BDDC method in PETSc for thermal and mechanical multi-material problems.

  Abstract
  In this work, we present a new variant of the balancing domain decomposition by constraints preconditioner that is robust for multi-material problems. We start with a well-balanced [...]

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• Simulation of high temperature superconductors and experimental validation

  M. Olm, S. Badia, A. Martín

  Computer Physics Communications (2019). Vol. 237, pp. 154-167

  
  

  Abstract

  In this work, we present a parallel, fully-distributed finite element numerical framework to simulate the low-frequency electromagnetic behaviour of superconducting devices, which efficiently exploits high performance computing platforms. We select the so-called H-formulation, which uses the magnetic field as a state variable. Nédélec elements (of arbitrary order) are required for an accurate approximation of the H-formulation for modelling electromagnetic fields along interfaces between regions with high contrast medium properties. An h-adaptive mesh refinement technique customized for Nédélec elements leads to a structured fine mesh in areas of interest whereas a smart coarsening is obtained in other regions. The composition of a tailored, robust, parallel nonlinear solver completes the exposition of the developed tools to tackle the problem. First, a comparison against experimental data is performed to show the availability of the finite element approximation to model the physical phenomena. Then, a selected state-of-the-art 3D benchmark is reproduced, focusing on the parallel performance of the algorithms. -

  Abstract
  In this work, we present a parallel, fully-distributed finite element numerical framework to simulate the low-frequency electromagnetic behaviour of superconducting [...]

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• Mixed aggregated finite element methods for the unfitted discretization of the Stokes problem

  S. Badia, A. Martín, F. Verdugo

  SIAM journal on scientific computing (2018). Vol. 40 (6), B1541-B1576

  
  

  Abstract

  In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that these kinds of methods lead to arbitrarily ill-conditioned systems and poorly approximated fluxes on unfitted interfaces/boundaries. In order to solve these issues, we consider the recently proposed aggregated finite element method, originally motivated for coercive problems. However, the well-posedness of the Stokes problem is far more subtle and relies on a discrete inf-sup condition. We consider mixed finite element methods that satisfy the discrete version of the inf-sup condition for body-fitted meshes and analyze how the discrete inf-sup is affected when considering the unfitted case. We propose different aggregated mixed finite element spaces combined with simple stabilization terms, which can include pressure jumps and/or cell residuals, to fix the potential deficiencies of the aggregated inf-sup. We carry out a complete numerical analysis, which includes stability, optimal a priori error estimates, and condition number bounds that are not affected by the small cut cell problem. For the sake of conciseness, we have restricted the analysis to hexahedral meshes and discontinuous pressure spaces. A thorough numerical experimentation bears out the numerical analysis. The aggregated mixed finite element method is ultimately applied to two problems with nontrivial geometries. No separate or additional fees are collected for access to or distribution of the work.

  Abstract
  In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that these kinds of methods lead to arbitrarily [...]

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• Robust and scalable domain decomposition solvers for unfitted finite element methods

  S. Badia, F. Verdugo

  Journal of Computational and Applied Mathematics (2018). Vol. 344, pp. 740-759

  
  

  Abstract

  Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since they avoid the generation of body-fitted meshes and the use of graph partitioning techniques, two main bottlenecks for problems with non-trivial geometries. However, the linear systems that arise from these discretizations can be much more ill-conditioned, due to the so-called small cut cell problem. The state-of-the-art approach is to rely on sparse direct methods, which have quadratic complexity and are thus not well suited for large scale simulations. In order to solve this situation, in this work we investigate the use of domain decomposition preconditioners (balancing domain decomposition by constraints) for unfitted methods. We observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior. As a result, we propose a customization of the classical BDDC methods based on the stiffness weighting operator and an improved definition of the coarse degrees of freedom in the definition of the preconditioner. These changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids. A complete set of complex 3D numerical experiments shows the good performance of the proposed preconditioners. -

  Abstract
  Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since [...]

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• Nonlinear parallel-in-time Schur complement solvers for ordinary differential equations

  S. Badia, M. Olm

  Journal of Computational and Applied Mathematics (2018). Vol. 344, pp. 794-806

  
  

  Abstract

  In this work, we propose a parallel-in-time solver for linear and nonlinear ordinary differential equations. The approach is based on an efficient multilevel solver of the Schur complement related to a multilevel time partition. For linear problems, the scheme leads to a fast direct method. Next, two different strategies for solving nonlinear ODEs are proposed. First, we consider a Newton method over the global nonlinear ODE, using the multilevel Schur complement solver at every nonlinear iteration. Second, we state the global nonlinear problem in terms of the nonlinear Schur complement (at an arbitrary level), and perform nonlinear iterations over it. Numerical experiments show that the proposed schemes are weakly scalable, i.e., we can efficiently exploit increasing computational resources to solve for more time steps the same problem. -

  Abstract
  In this work, we propose a parallel-in-time solver for linear and nonlinear ordinary differential equations. The approach is based on an efficient multilevel solver [...]

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• The aggregated unfitted finite element method for elliptic problems

  S. Badia, F. Verdugo, A. Martín

  Computer Methods in Applied Mechanics and Engineering (2018). Vol. 336, pp. 533-553

  
  

  Abstract

  Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hindersthe practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.

  Abstract
  Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone [...]

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• Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes

  S. Badia, J. Bonilla, A. Hierro

  Computer Methods in Applied Mechanics and Engineering (2017). Vol. 320, pp. 582-605 (Preprint)

  
  

  Abstract

  This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport and convection–diffusion problems and the respective transient problems with implicit time integration. Monotonic schemes that combine explicit time stepping with dG space discretization are very common, but the design of such schemes for implicit time stepping is rare, and it had only been attained so far for 1D problems. The proposed scheme is based on a piecewise linear dG discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas. In order to define the new shock detector, we have introduced the concept of discrete local extrema. The diffusion operator is a graph-Laplacian, instead of the more common finite element discretization of the Laplacian operator, which is essential to keep monotonicity on general meshes and in multi-dimension. The resulting nonlinear stabilization is non-smooth and nonlinear solvers can fail to converge. As a result, we propose a smoothed (twice differentiable) version of the nonlinear stabilization, which allows us to use Newton with line search nonlinear solvers and dramatically improve nonlinear convergence. A theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties. Further, the resulting operator is Lipschitz continuous and there exists at least one solution of the discrete problem, even in the non-smooth version. We provide a set of numerical results to support our findings.

  Abstract
  This work is devoted to the design of interior penalty discontinuous Galerkin (dG) schemes that preserve maximum principles at the discrete level for the steady transport [...]

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• Convergence to suitable weak solutions for a finite element approximation of the Navier–Stokes equations with numerical subgrid scale modeling

  S. Badia, J. Gutiérrez-Santacreu

  Journal of Scientific Computing (2017). Vol. 71 (1), pp. 386-413 (Preprint)

  
  

  Abstract

  In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider a subgrid model that enforces orthogonality between subgrid and finite element components. Further, the subgrid component must be tracked in time. Since this type of schemes introduce pressure stabilization, we have proved the result for equal-order velocity and pressure finite element spaces that do not satisfy a discrete inf-sup condition

  Abstract
  In this work we prove that weak solutions constructed by a variational multiscale method are suitable in the sense of Scheffer. In order to prove this result, we consider [...]

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• Space-time balancing domain decomposition

  S. Badia, M. Olm

  SIAM journal on scientific computing (2017). Vol. 39 (2), pp. 194-213 (Preprint)

  
  

  Abstract

  In this work, we propose two-level space-time domain decomposition preconditioners for parabolic problems discretized using finite elements. They are motivated as an extension to space-time of balancing domain decomposition by constraints preconditioners. The key ingredients to be defined are the subassembled space and operator, the coarse degrees of freedom (DOFs) in which we want to enforce continuity among subdomains at the preconditioner level, and the transfer operator from the subassembled to the original finite element space. With regard to the subassembled operator, a perturbation of the time derivative is needed to end up with a well-posed preconditioner. The set of coarse DOFs includes the time average (at the space-time subdomain) of classical space constraints plus new constraints between consecutive subdomains in time. Numerical experiments show that the proposed schemes are weakly scalable in time, i.e., we can efficiently exploit increasing computational resources to solve more time steps in the same total elapsed time. Further, the scheme is also weakly space-time scalable, since it leads to asymptotically constant iterations when solving larger problems both in space and time. Excellent wall clock time weak scalability is achieved for space-time parallel solvers on some thousands of cores No separate or additional fees are collected for access to or distribution of the work. -

  Abstract
  In this work, we propose two-level space-time domain decomposition preconditioners for parabolic problems discretized using finite elements. They are motivated as an [...]

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