Mixed finite element method

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In numerical analysis, the mixed finite element method, is a type of finite element method in which extra fields to be solved are introduced during the posing a partial differential equation problem. Somewhat related is the hybrid finite element method. The extra fields are constrained by using Lagrange multiplier fields. To be distinguished from the mixed finite element method, usual finite element methods that do not introduce such extra fields are also called irreducible or primal finite element methods. [1] The mixed finite element method is efficient for some problems that would be numerically ill-posed if discretized by using the irreducible finite element method; one example of such problems is to compute the stress and strain fields in an almost incompressible elastic body.

In mixed methods, the Lagrange multiplier fields inside the elements, usually enforcing the applicable partial differential equations. [2] This results in a saddle point system having negative pivots and eigenvalues, rendering the system matrix to be non-positive which results in complications in solving for it. In sparse direct solvers, pivoting may be needed, where ultimately the resulting matrix has 2x2 blocks on the diagonal, [3] rather than a working towards a completely pure LLH Cholesky decomposition for positive definite symmetric or Hermitian systems. Pivoting may result in unpredictable memory usage increases. For iterative solvers, only GMRES based solvers work, rather than slightly "cheaper" MINRES based solvers.

In hybrid methods, the Lagrange fields are for jumps of fields between elements, living on the boundary of the elements, weakly enforcing continuity; continuity from fields in the elements does not need to be enforced through shared degrees of freedom between elements anymore. Both mixing and hybridization can be applied simultaneously. [4] These enforcements are "weak", i.e. occur upon having the solutions and possibly only at some points or e.g. matching moment integral conditions, rather than "strong" in which case the conditions are fulfilled directly in the type of solutions sought. Apart from the harmonics (non-trivial solution to the homogeneous equations at zero loads), hybridization allows for static Guyan condensation of the discontinuous fields internal to the elements, reducing the number of degrees of freedom, and moreover reducing or eliminating the number of negative eigenvalues and pivots resulting from application of the mixed method.

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Domain decomposition methods

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The finite element method (FEM) is a widely used method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables. To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.

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In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.

GetFEM++

GetFEM++ is a generic finite element C++ library with interfaces for Python, Matlab and Scilab. It aims at providing finite element methods and elementary matrix computations for solving linear and non-linear problems numerically. Its flexibility in choosing among different finite element approximations and numerical integration methods is one of its distinguishing characteristics.

References

  1. ↑ Olek C Zienkiewicz, Robert L Taylor and J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Elsevier.
  2. ↑ Arnold, Douglas. "Douglas Arnold 2016 Woudschoten Conference" (PDF).
  3. ↑ "Sparse block factorization of saddle point matrices". Linear Algebra and its Applications. 502: 214–242. 2016-08-01. doi: 10.1016/j.laa.2015.07.042 . ISSN   0024-3795.
  4. ↑ "2019 Feb 1, Bernardo Cockburn, University of Minnesota, Variational principles for hybridizable discontinuous Galerkin methods: A short story". PSU Media Space. Retrieved 2021-05-02.


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