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On a Non-Uniform $α$-Robust IMEX-L1 Mixed FEM for Time-Fractional PIDEs

Lok Pati Tripathi, Aditi Tomar, Amiya K. Pani

Abstract

A non-uniform implicit-explicit L1 mixed finite element method (IMEX-L1-MFEM) is investigated for a class of time-fractional partial integro-differential equations (PIDEs) with space-time dependent coefficients and non-self-adjoint elliptic part. The proposed fully discrete method combines an IMEX-L1 method on a graded mesh in the temporal variable with a mixed finite element method in spatial variables. The focus of the study is to analyze stability results and to establish optimal error estimates, up to a logarithmic factor, for both the solution and the flux in $L^2$-norm when the initial data $u_0\in H_0^1(Ω)\cap H^2(Ω)$. Additionally, an error estimate in $L^\infty$-norm is derived for 2D problems. All the derived estimates and bounds in this article remain valid as $α\to 1^{-}$, where $α$ is the order of the Caputo fractional derivative. Finally, the results of several numerical experiments conducted at the end of this paper are confirming our theoretical findings.

On a Non-Uniform $α$-Robust IMEX-L1 Mixed FEM for Time-Fractional PIDEs

Abstract

A non-uniform implicit-explicit L1 mixed finite element method (IMEX-L1-MFEM) is investigated for a class of time-fractional partial integro-differential equations (PIDEs) with space-time dependent coefficients and non-self-adjoint elliptic part. The proposed fully discrete method combines an IMEX-L1 method on a graded mesh in the temporal variable with a mixed finite element method in spatial variables. The focus of the study is to analyze stability results and to establish optimal error estimates, up to a logarithmic factor, for both the solution and the flux in -norm when the initial data . Additionally, an error estimate in -norm is derived for 2D problems. All the derived estimates and bounds in this article remain valid as , where is the order of the Caputo fractional derivative. Finally, the results of several numerical experiments conducted at the end of this paper are confirming our theoretical findings.

Paper Structure

This paper contains 8 sections, 15 theorems, 135 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Mixed formulation and some useful results
  3. Non-uniform IMEX-L1-MFEM
  4. Stability Analysis
  5. Error analysis
  6. Numerical results
  7. Conclusion
  8. Proof of Lemma \ref{['dfdoi']}

Key Result

Theorem 2.1

Let $f \in W^{3,1}(J;L^2(\Omega))\cap W^{2,1}(J;H^1(\Omega))$, $u_0 \in \dot{H}^2(\Omega)=H_0^1(\Omega)\cap H^2(\Omega)$, and for $k=0,1,2,3,$ for some positive constants $C_{\mathcal{I}}$, $C_{\boldsymbol{A}}$, $C_{\boldsymbol{b}}$ and $C_c$. Then, the problem (pide) is well-posed, and its solution $u$ satisfies the following regularity results for some positive constant $C$ which remains bound

Theorems & Definitions (46)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.1
  • Remark 3.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 36 more