A POSTERIORI ERROR ESTIMATES FOR APPROXIMATE SOLUTIONS OF THE OBSTACLE PROBLEM FOR THE

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Abstract

The paper is concerned with a functional identity and estimates which are fulfilled for the measures of deviations from exact solutions of the obstacle problem for the

About the authors

D. E Apushkinskaya

People’s Friendship University of Russia named after Patrice Lumumba

Email: apushkinskaya@gmail.com
Moscow, Russia

A. A Novikova

People’s Friendship University of Russia named after Patrice Lumumba

Email: aanovikova01@gmail.com
Moscow, Russia

S. I Repin

People’s Friendship University of Russia named after Patrice Lumumba; Saint Petersburg Department of Steklov Mathematical Institute of RAS

Email: rpnspb@gmail.com
Moscow, Russia

References

  1. Lions, J.-L. Variational inequalities / J.-L. Lions, G. Stampacchia // Comm. Pure Appl. Math. — 1967. — V. 20. — P. 493-519.
  2. Petrosyan, A. Regularity of Free Boundaries in Obstacle-Type Problems / A. Petrosyan, H. Shagholian, N.N. Uraltseva. — Providence : American Mathematical Society, 2012. — 221 p.
  3. Choe, H.J. On the obstacle problem for quasilinear elliptic equations of р-Laplacian type / H.J. Choe, J.L. Lewis // SIAM J. Math. Anal. — 1991. — V. 22, № 3. — P. 623-638.
  4. Andersson J. Optimal regularity for the obstacle problem for the р-Laplacian / J. Andersson, E. Lindgren, H. Shahgholian // J. Differ. Equat. — 2015. — V. 259, № 6. — P. 2167-2179.
  5. Jouvet, G. Steady, shallow ice sheets as obstacle problems: well-posedness and finite element approximation / G. Jouvet, E. Bueler // SIAM J. Appl. Math. — 2012. — V. 72, № 4. — P. 1292-1314.
  6. Lewicka, M. The obstacle problem for the р-Laplacian via optimal stopping of tug-of-war games / M. Lewicka, J.J. Manfredi // Probab. Theory Related Fields. — 2017. — V. 167, № 1-2. — P. 349-378.
  7. On the porosity of free boundaries in degenerate variational inequalities / L. Karp, T. Kipelainen, A. Petrosyan, H. Shagholian // J. Differ. Equat. — 2000. — V. 164, № 1. — P. 110-117.
  8. Lee, K. Hausdorff measure and stability for the p-obstacle problem (2
  9. Rodrigues, J.F. Stability remarks to the obstacle problem for p-Laplacian type equations / J.F. Rodrigues // Calc. Var. Partial Differ. Equat. — 2005. — V. 23, № 1. — P. 51-65.
  10. Falk, R.S. Error estimates for approximation of a class of a variational inequalities / R.S. Falk // Math. Comp. — 1974. — V. 28. — P. 963-971.
  11. Chen, Z., Residual type a posteriori error estimates for elliptic obstacle problems / Z. Chen, R. Nochetto // Numer. Math. — 2000. — V. 84. — P. 527—548.
  12. Repin, S.I. Accuracy of Mathematical Models — Dimension Reduction, Homogenization, and Simplification / S.I. Repin, S.A. Sauter. — Zurich : European Mathematical Society, 2020. — 317 p.
  13. Repin, S.I. A posteriori error estimation for variational problems with uniformly convex functionals / S.I. Repin // Math. Comp. — 2000. — V. 69, № 230. — P. 481-500.
  14. Репин, С.И. Апостериорные тождества для мер отклонений от точных решений нелинейных краевых задач / С.И. Репин // Журн. вычислит. математики и мат. физики. — 2023. — Т. 63, № 6. — С. 896-919.
  15. Repin, S. Error identities for variational problems with obstacles / S. Repin, J. Valdman // ZAMM Z. Angew. Math. Mech. — 2018. — Bd. 98, № 4. — S. 635-658.
  16. Apushkinskaya D.E. Thin obstacle problem: estimates of the distance to the exact solution / D.E. Apushkinskaya, S.I. Repin // Interfaces Free Bound. — 2018. — V. 20, № 4. — P. 511-531.
  17. Apushkinskaya, D.E. and Repin, S.I., Biharmonic obstacle problem: guaranteed and computable error bounds for approximate solutions, Comput. Math. Math. Phys., 2020, vol. 60, no. 11, pp. 1823-1838.
  18. Apushkinskaya, D. Functional a posteriori error estimates for the parabolic obstacle problem / D. Apushkinskaya, S. Repin // Comput. Methods Appl. Math. — 2022. — V. 22, № 2. — P. 259276.
  19. Sharp numerical inclusion of the best constant for embedding H01 (Q) < Lp(Q) on bounded convex domain / K. Tanaka, K. Sekine, M. Mizuguchi, S. Oishi // J. Comput. Appl. Math. — 2017. — V. 311 — P. 306-313.
  20. Rossi, J.D. Optimal regularity at the free boundary for the infinity obstacle problem / J.D. Rossi, E.V. Teixeira, J.V. Urbano // Interfaces Free Bound. — 2015. — V. 17, № 3. — P. 381-398.

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