A New Discretization for mth-Laplace Equations with Arbitrary Polynomial Degrees

This paper introduces new mixed formulations and discretizations for $m$th-Laplace equations of the form $(-1)^m\Delta^m u=f$ for arbitrary $m=1,2,3,\dots$ based on novel Helmholtz-type decompositions for tensor-valued functions. The new discretizations allow for ansatz spaces of arbitrary polynomial degree and the lowest-order choice coincides with the non-conforming FEMs of Crouzeix and Raviart for $m=1$ and of Morley for $m=2$. Since the derivatives are directly approximated, the lowest-order discretizations consist of piecewise affine and piecewise constant functions for any $m=1,2,\dots$ Moreover, a uniform implementation for arbitrary $m$ is possible. Besides the a priori and a posteriori analysis, this paper proves optimal convergence rates for adaptive algorithms for the new discretizations.

[1]  ROB STEVENSON,et al.  The completion of locally refined simplicial partitions created by bisection , 2008, Math. Comput..

[2]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..

[3]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[4]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[5]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[6]  A. Ženíšek.  Interpolation polynomials on the triangle , 1970 .

[7]  Rolf Stenberg,et al.  A posteriori error estimates for the Morley plate bending element , 2007, Numerische Mathematik.

[8]  Mira Schedensack,et al.  A New Generalization of the P 1 Non-Conforming FEM to Higher Polynomial Degrees , 2015, Comput. Methods Appl. Math..

[9]  W. Rudin.  Principles of mathematical analysis , 1964 .

[10]  Ming Wang,et al.  Minimal finite element spaces for 2m-th-order partial differential equations in Rn , 2012, Math. Comput..

[11]  Mira Schedensack,et al.  Mixed finite element methods for linear elasticity and the Stokes equations based on the Helmholtz decomposition , 2017 .

[12]  Dietmar Gallistl,et al.  Stable splitting of polyharmonic operators by generalized Stokes systems , 2017, Math. Comput..

[13]  Andreas Veeser,et al.  Approximating Gradients with Continuous Piecewise Polynomial Functions , 2014, Found. Comput. Math..

[14]  Jun Hu,et al.  A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes , 2014, Comput. Math. Appl..

[15]  D. Arnold,et al.  A uniformly accurate finite element method for the Reissner-Mindlin plate , 1989 .

[16]  John R. King,et al.  The isolation oxidation of silicon: the reaction-controlled case , 1989 .

[17]  Carsten Carstensen,et al.  Axioms of adaptivity for separate marking , 2016, 1606.02165.

[18]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[19]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[20]  S. C. Brenner,et al.  C 0 Interior Penalty Methods , 2011 .

[21]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[22]  John W. Barrett,et al.  Finite element approximation of a sixth order nonlinear degenerate parabolic equation , 2004, Numerische Mathematik.

[23]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[24]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[25]  T. Hughes,et al.  Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity , 2002 .

[26]  Rüdiger Verführt,et al.  A review of a posteriori error estimation and adaptive mesh-refinement techniques , 1996, Advances in numerical mathematics.

[27]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[28]  Thirupathi Gudi,et al.  An interior penalty method for a sixth-order elliptic equation , 2011 .

[29]  L. Morley.  The Triangular Equilibrium Element in the Solution of Plate Bending Problems , 1968 .

[30]  P. Raviart,et al.  Conforming and nonconforming finite element methods for solving the stationary Stokes equations I , 1973 .

[31]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[32]  Necas Jindrich.  Les Méthodes directes en théorie des équations elliptiques , 2017 .

[33]  Ronald A. DeVore,et al.  Fast computation in adaptive tree approximation , 2004, Numerische Mathematik.