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A new expanded mixed element method for convection-dominated Sobolev equation.

Jinfeng Wang1, Yang Liu2, Hong Li2

  • 1School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China.

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Summary
This summary is machine-generated.

A novel expanded mixed element method is introduced for convection-dominated Sobolev equations. This method offers optimal error estimates for scalar unknowns and their gradients, demonstrating improved efficiency in numerical simulations.

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • The classical expanded mixed element method uses H(div; Ω) space for gradients.
  • Convection-dominated Sobolev equations pose numerical challenges.

Purpose of the Study:

  • To propose and analyze a new expanded mixed element method.
  • To address challenges in solving convection-dominated Sobolev equations.
  • To establish theoretical guarantees for the new method's solutions.

Main Methods:

  • Developed a new expanded mixed element method with gradients in a simpler square-integrable space.
  • Proved existence and uniqueness of the finite element solution.
  • Introduced a new expanded mixed projection.
  • Derived optimal a priori error estimates in L(2) and H(1) norms.

Main Results:

  • Established existence and uniqueness for the finite element solution.
  • Derived optimal a priori error estimates in L(2)-norm for the scalar unknown u.
  • Obtained a priori error estimates in (L(2))^2-norm for the gradient λ and flux σ.
  • Achieved optimal a priori error estimates in H(1)-norm for the scalar unknown u.
  • Numerical results validated the method's efficiency.

Conclusions:

  • The new expanded mixed element method is effective for convection-dominated Sobolev equations.
  • The method provides optimal error estimates, enhancing accuracy.
  • Numerical experiments confirm the practical efficiency of the proposed technique.