Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates

1998/01/01 by Alfred Schatz, A. H. Schatz · 5 citations
Engineering · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Numerical methods in engineering

paper · pdf · doi:10.1090/s0025-5718-98-00959-4

Abstract

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb R^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In a sense to be discussed below these sharpen known quasi–optimal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_∞ </mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript normal infinity Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">W¹_∞ </mml:annotation> </mml:semantics> </mml:math> </inline-formula> estimates for the error on irregular quasi–uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u"> <mml:semantics> <mml:mi>u</mml:mi> <mml:annotation encoding="application/x-tex">u</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript upper N"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb R^N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non–smooth problems.

Cited by

• Pointwise error estimates of linear finite element method for Neumann boundary value problems in a smooth domain
• The pointwise stabilities of piecewise linear finite element method on non-obtuse tetrahedral meshes of nonconvex polyhedra
• C⁰ discontinuous Galerkin finite element methods for second order linear elliptic partial differential equations in non-divergence form
• Numerical analysis of two Galerkin discretizations with graded temporal grids for fractional evolution equations
• L^∞- and W^1,∞-error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Related

• Interior maximum norm estimates for finite element methods
• Local energy estimates for the finite element method on sharply varying grids
• Finite Element Error Estimates for Critical Growth Semilinear Problems without Angle Conditions
• Adaptive finite element approximations for elliptic problems using regularized forcing data
• A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds