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A space-time finite element method for parabolic obstacle problems

José Joaquín Carvajal, Davood Damircheli, Thomas Führer, Francisco Fuica, Michael Karkulik

Abstract

We propose and analyze a general framework for space-time finite element methods that is based on least-squares finite element methods for solving a first-order reformulation of the thick parabolic obstacle problem. Discretizations based on simplicial and prismatic meshes are studied and we show a priori error estimates for both. Convergence rates are derived for sufficiently smooth solutions. Reliable a posteriori bounds are provided and used to steer an adaptive algorithm. Numerical experiments including a one-phase Stefan problem and an American option pricing problem are presented.

A space-time finite element method for parabolic obstacle problems

Abstract

We propose and analyze a general framework for space-time finite element methods that is based on least-squares finite element methods for solving a first-order reformulation of the thick parabolic obstacle problem. Discretizations based on simplicial and prismatic meshes are studied and we show a priori error estimates for both. Convergence rates are derived for sufficiently smooth solutions. Reliable a posteriori bounds are provided and used to steer an adaptive algorithm. Numerical experiments including a one-phase Stefan problem and an American option pricing problem are presented.

Paper Structure

This paper contains 20 sections, 12 theorems, 115 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. First-order formulation and least-squares discretization
  4. Sobolev and Bochner--Sobolev spaces
  5. Well-posedness of canonic variational formulation
  6. First-order system and variational formulation
  7. Numerical method
  8. Simplicial meshes
  9. Prismatic meshes
  10. A priori and a posteriori analysis
  11. Quasi-interpolators for analysis on tensor mesh
  12. A priori error analysis
  13. Convergence rates for Lagrange finite element discretization
  14. Convergence rates for Gantner--Stevenson finite element discretization
  15. A posteriori error estimator
  16. ...and 5 more sections

Key Result

Theorem 1

Space $\mathcal{W}$ is continuously embedded in the space $C^0([0,T];L^{2}(\Omega))$. Moreover, the integration by parts formula holds true for all $v, w \in \mathcal{W}$.

Figures (6)

  • Figure 1: Stefan problem
  • Figure 2: Pyramide obstacle
  • Figure 3: Left column shows simplicial meshes generated by adaptive algorithm for the problem from Section \ref{['sec:numerics:pyramide']} (vertical axis corresponds to time). Right column shows solution component $u_{\mathcal{P}}$.
  • Figure 4: American option problem from section \ref{['sec:blackscholes']} on a sequence of uniformly (top left) and adaptively (top right) refined meshes. Bottom plot compares a specific estimator contribution.
  • Figure 5: Four consecutive meshes generated by the adaptive algorithm for the American option pricing problem. Vertical axis corresponds to time.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Theorem 1: embedding, integration by parts
  • Theorem 2: CharrierTroianello78
  • Remark 3
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • ...and 16 more