Abstract
A new positive definite expanded mixed finite element method is proposed for parabolic partial integrodifferential equations. Compared to expanded mixed scheme, the new expanded mixed element system is symmetric positive definite and both the gradient equation and the flux equation are separated from its scalar unknown equation. The existence and uniqueness for semidiscrete scheme are proved and error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are provided to confirm our theoretical analysis.
Highlights
In this paper, we consider the following initial-boundary value problem of parabolic partial integrodifferential equations:t ut − ∇ · a x, t ∇u b x, t ∇uds f x, t, x, t ∈ Ω × J, 1.1 u x, t 0, x, t ∈ ∂Ω × J, u x, 0 u0 x, x ∈ Ω, where Ω is a bounded convex polygonal domain in Rd, d 1, 2, 3 with a smooth boundary ∂Ω, J 0, T is the time interval with 0 < T < ∞
Compared to standard mixed methods whose numerical solutions have been quite difficult because of losing positive definite properties, the proposed one does not lead to some saddle point problems
Our purpose is to propose and analyse a new expanded mixed method based on the positive definite system 32–34 for parabolic integrodifferential equations
Summary
We consider the following initial-boundary value problem of parabolic partial integrodifferential equations:. In 1994, Chen 11, proposed a new mixed method, which is called a expanded mixed finite element method and proved some mathematical theories for second-order linear elliptic equation. Compared to standard mixed methods whose numerical solutions have been quite difficult because of losing positive definite properties, the proposed one does not lead to some saddle point problems. Our purpose is to propose and analyse a new expanded mixed method based on the positive definite system 32–34 for parabolic integrodifferential equations. Compared to expanded mixed methods, the proposed mixed element system is symmetric positive definite and avoids some saddle point problems. What is more, both the gradient equation and the flux equation are separated from its scalar unknown equation. With norm · L2 Ω or · L2 Ω and introduce the function space W H div; Ω L2 Ω d; ∇ · ω ∈ L2 Ω }
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