Space-time least-squares finite elements for parabolic equations

@article{Fhrer2019SpacetimeLF,
  title={Space-time least-squares finite elements for parabolic equations},
  author={Thomas F{\"u}hrer and Michael Karkulik},
  journal={Comput. Math. Appl.},
  year={2019},
  volume={92},
  pages={27-36}
}
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26 Citations
Background Citations
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Methods Citations
3

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26 Citations

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References

SHOWING 1-10 OF 40 REFERENCES

Space-Time Finite Element Methods for Parabolic Problems

  • O. Steinbach
  • Mathematics, Computer Science
    Comput. Methods Appl. Math.
  • 2015
This approach allows the use of general and unstructured space-time finite elements which do not require any tensor product structure and the stability of the numerical scheme is based on a stability condition which holds for standard finite element spaces.

Coercive space-time finite element methods for initial boundary value problems

Abstract. We propose and analyse new space-time Galerkin-Bubnov-type finite element formulations of parabolic and hyperbolic second-order partial differential equations in finite time intervals.…

Space–time hp-approximation of parabolic equations

A new space–time finite element method for the solution of parabolic partial differential equations is introduced. In a mesh and degree-dependent norm, it is first shown that the discrete bilinear…

Space-Time Adaptive Methods for the Mixed Formulation of a Linear Parabolic Problem

Numerical results for backward Euler and CN schemes are presented to compare their performance in the time adaptivity setting over uniform/adaptive spatial meshes.

Least-Squares Galerkin Methods for Parabolic Problems II: The Fully Discrete Case and Adaptive Algorithms

This second part of this series on least-squares Galerkin methods for parabolic initial-boundary value problems studies the full discretization in time and space and proves a strengthened Cauchy--Schwarz inequality between coarse and hierarchical surplus space and a bound on the condition number of the auxiliary problem.

Space-time discretizations using constrained first-order system least squares (CFOSLS)

A stable space–time finite element method for parabolic evolution problems

This paper is concerned with the analysis of a new stable space–time finite element method (FEM) for the numerical solution of parabolic evolution problems in moving spatial computational domains.…

New a priori analysis of first‐order system least‐squares finite element methods for parabolic problems

We provide new insights into the a priori theory for a time‐stepping scheme based on least‐squares finite element methods for parabolic first‐order systems. The elliptic part of the problem is of…

Space-time least-squares isogeometric method and efficient solver for parabolic problems

A space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space- time domain and well-suited for parallelization is proposed.

Least-Squares Galerkin Methods for Parabolic Problems I: Semidiscretization in Time

The derivation and analysis of one-step methods for semi-discretization in time from least-squares principles for linear parabolic problems with specific combination of piecewise linear, not necessarily continuous, functions with continuous piecewiselinear functions for the flux and scalar variable, respectively.