A quasioptimal estimate in piecewise polynomial Galerkin approximation of parabolic problems

1982 ◽  
pp. 230-245 ◽  
Author(s):  
Lars B. Wahlbin
Keyword(s):  
Galerkin Approximation ◽  
Parabolic Problems ◽  
Piecewise Polynomial

scholarly journals Uniform stability of linear multistep methods in Galerkin procedures for parabolic problems

1979 ◽  
Vol 2 (4) ◽  
pp. 651-667 ◽  
Author(s):  
Eckart Gekeler
Keyword(s):  
Stability Region ◽  
Multistep Methods ◽  
Elliptic Boundary ◽  
Galerkin Approximation ◽  
Parabolic Problems ◽  
Linear Multistep Methods ◽  
Spectral Condition ◽  
Initial Value ◽  
Time Step ◽  
Elliptic Boundary Value

Linear multistep methods are considered which have a stability regionSand areD-stable on the whole boundary∂S⊂S of S. Error estimates are derived which hold uniformly for the class of initial value problemsY′=AY+B(t),t>0,Y(0)=Y0with normal matrixAsatisfying the spectral conditionSp(ΔtA)⊂S,Δttime step,Sp(A)spectrum ofA. Because of this property, the result can be applied to semidiscrete systems arising in the Galerkin approximation of parabolic problems. Using known results of the Ritz theory in elliptic boundary value problems error bounds for Galerkin multistep procedures are then obtained in this way.

scholarly journals Simulation of continuously deforming parabolic problems by Galerkin finite-elements method

1986 ◽  
Vol 9 (3) ◽  
pp. 577-582
Author(s):  
Yahia S. Halabi
Keyword(s):  
Phase Change ◽  
Hermite Polynomials ◽  
Galerkin Approximation ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Two Phase ◽  
Transition Zones ◽  
Basic Functions ◽  
Numerical Finite Element

A general numerical finite element scheme is described for parabolic problems with phase change wherein the elements of the domain are allowed to deform continuously. The scheme is based on the Galerkin approximation in space, and finite difference approximation for the time derivatives. The numerical scheme is applied to the two-phase Stefan problems associated with the melting and solidification of a substance. Basic functions based on Hermite polynomials are used to allow exact specification of flux-latent heat balance conditions at the phase boundary. Numerical results obtained by this scheme indicates that the method is stable and produces an accurate solutions for the heat conduction problems with phase change; even when large time steps used. The method is quite general and applicable for a variety of problems involving transition zones and deforming regions, and can be applied for one multidimensional problems.

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