The Convergence of Discretization Methods if Applied to Weakly Formulated Problems. Theory and Examples

Abstract

Dedicated to the memory of LOTHAR COLLATZ Die von STUMMEL und GRIGORIEFF begriindete und spater von zahlreichen Autoren auf nichtlineare Probleme erweiterte allgemeine Theorie der Konvergenz von Diskretisierungsalgorithmen geniigt nicht vollstandig der numerischen Behandlung hearer oder nichtlinearer, schwach formulierter Probleme. Eine derartige Erweiterung wird hier zusammen mit Beispielen fur nichtlineare elliptische und hyperbolische partielle Differentialgleichungen angegeben. Oft fiihrt der Ubergang von klassischen zu schwachen Losungen zu einem Verlust der Eindeutigkeit, so daJ Eindeutigkeit nur durch zusatzliche Bedingungen wie etwa geeignete Ungleichungsrestriktionen garantiert werden kann. Wie solche Restriktionen im allgemeinen diskretisiert werden miissen, um eine Konvergenz der numerischen Losung gegen die entsprechende eindeutige schwache Losung zu erreichen, wird ebenfalls diskutiert, und die Ergebnisse werden insbesondere mit den Entropie-Bedingungen von Erhaltungssatzen verglichen. Die Beispiele werden nur behandelt, um die allgemeine Theorie zu motivieren und deren Anwendbarkeit zu demonstrieren. The general theory of the convergence of discretization algorithms founded by STUMMEL and GRIGORIEFF and later extended to nonlinear problems by numerous authors does not completely fit the numerical treatment of linear or nonlinear weakly formulated problems. Such an extension is presented together with examples of nonlinear elliptic and hyperbolic PDEs. Often the transition from classsical to weak solutions leads to a loss of uniqueness such that uniqueness can only be guaranteed by additional conditions, e.g. suitable inequality constraints. How such constraints must generally be discretized in order to make the numerical solution converge to the corresponding unique weak solution is also discussed, and the results are particularly compared with entropy conditions of conservation laws. The examples are only treated in order to motivate the general theory and to demonstrate its applicability. OcHosaHHan CTYMMEJIEM u ~PMI-OPMEBOM u pacutupartnasl n o 3~e MnozorlucneHHHbiuu asmopaMu Ha cnyrlaii nenuHeiiHbix 3adar o6u4an meopusl cxoduuocmu aflropumMo8 dyc~pemu3ayuu necoseputenHo coomsemcmsyem rlucnenftoii o6pa6om~e nuHeiiHbrx UAU cna60 @opuynuposanHbix Henuneiimix 3adav. Jaemcsl mauoe pacutupewe sMecme c npuuepauu dnsl HefluHeiinbix 3 n f l~n m~r l e c~~x u zunep6onuuec~ux ypasnenuii 6 VacmHbix npou36odnbix. Liacmo nepexod om KnaccurlecKux Ha cfla6bie peutewisl sedem Ha nomepm eduttcmsemocmu maK, rlmo M O X H O 06ecnerlusamb eduncmsennocmb monbKo npu dononnumenbitbix ycnosuslx, Ha npuuep npu nodxodnupx ycilosuslx s aude HepaseHcms. M3yrlaemcsl u sonpoc, Kau 8006tye mame ycnosun Hado duc~pemu3uposam~ rlrno6bi 06ecnevusamb cxoduuocmb Y U C A~H~O Z O peuteHus K coomsemcmsymtqeuy eduncmsennouy cna6ouy peweHum u cpasnslmmcn pe3ynbmambr 8 o c o 6 e~~o c m u u ycilosusluu snmponuu 3a~onos coxpaHeHun. IIpuMepbr monbuo u3yrlammcn 6 uavecmse uomusayuu o6u4eli meopuu u no~a3amenbcmsa ezo npuueltuwocmu. MSC (1980): ...

A Discretization Theory for Monotone Semicoercive Problems and Finite Element Convergence for p‐Harmonic Signorini Problems

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“…Moreover, according to the properties of subdifferentials, we have the following r e s u l t : If the functional Q(u) has a subdifferential at u E H ( H is a Hilbert space), then this subdifferential i s a convex closed set; hence for u E D ( K ) , the set Ku is convex and closed. Now we show that {el(B +-K ) } is pointwise convergent to B + K , i.e., ~n E Ei(B + K ) 71 7 Yn -+ YO * YO E ( B + K ) 11,. In fact, we can express y , as wn + v, where w, = P,Bu, v, E P,Ku. Because of P, -+ I , hence w, -+ wo = Bu.…”

“…This paper is based on discretization theory and can be regarded as a continuation of investigations in [ 2 ] , [7], [5]. In Sections 2 and 3, we therefore generalize at first some results for one-valued operators in [2], [7], [5] to the case of setvalued operators. Then, as an application, two examples for set-valued differential equations are given in Section 4.…”

Nonlinear demi-regular approximation solvability of equations involving strongly accretive operators

“…Ansorge and Lei [4] studied problems on the solvability of nonlinear elliptic equations and nonlinear hyperbolic equations with interesting examples. More significantly, Anselone and Lei [3] generalized the regular operator approximation theory to the case of the demiregular convergence and applied the obtained results to the case of the fixed point and bifurcation approximations with examples.…”