The finite element method explicit scheme for a solution of one problem of surface and ground water combined movement

Transcription

1 IOP Conference Seres: Materals Scence and Engneerng PAPER OPEN ACCESS e fnte element metod explct sceme for a soluton of one problem of surface and ground water combned movement o cte ts artcle: L L Glazyrna and M F Pavlova 6 IOP Conf. Ser.: Mater. Sc. Eng Related content - On surface electromagnetc waves V N Datsko and A A Kopylov - An Explct Sceme for te KdV Equaton Wang Hu-Png Wang Yu-Sun and Hu Yng-Yng - A new parallelzaton algortm of ocean model wt explct sceme X D Fu Vew te artcle onlne for updates and enancements. s content was downloaded from IP address on 7//7 at 3:5

2 t Internatonal Conference on "Mes metods for boundary-value problems and applcatons" IOP Publsng IOP Conf. Seres: Materals Scence and Engneerng 58 (6) 4 do:.88/ x/58//4 e fnte element metod explct sceme for a soluton of one problem of surface and ground water combned movement L L Glazyrna and M F Pavlova Kazan Federal Unversty 8 Kremlyovskaya str. Kazan 48 Russa E-mal: glazyrna-ludmla@ya.ru Abstract. We nvestgate te ntal-boundary value problem for te system of two nonlnear degenerate parabolc combned equatons wen te one of equatons s set n te doman we consder and te second s set on te conducted secton of ts doman. Wt elp of te semdscretzaton metods wt respect to tme varable and fnte element metod (FEM) wt respect to space varable we construct an explct dfference sceme. We obtaned condtons wc provde us convergence of constructed metod. e result of convergence was obtaned wt te mnmum condtons for smootness of te ntal data.. Statement of te problem s paper s devoted to te convergence nvestgaton of an approxmate soluton metod for te followng ntal-boundary problem: ( u) ( x t) ( k ( x u( x t) u( x t) )) f ( x t) x \ t ( ) t x () ( u) u ( x t ) k x u ( x t ) ( x t ) t s s ( ( )) [ k( x u( x t) u( x t) ) cos( n x) ] f( x t) () [ u] x t ( ) u( x) u ( x) x; u( x t) x t ( ). (3) Here s a bounded doman n space R s ts boundary s a conducted secton of and dvdng t nto two connected domans [ ] s a dscontnuty of a functon wen ts passng troug te secton n s a normal to. Problem () (3) descrbes (see []) te process of surface-water and groundwater jont moton n addton te secton s a bed rver or cannel te requred functon u defnes te egt of te water face above te mpenetrable base. We assume tat functons ( ) are absolutely contnuous strongly ncreasng and satsfy te followng condtons for arbtrary R Content from ts work may be used under te terms of te Creatve Commons Attrbuton 3. lcence. Any furter dstrbuton of ts work must mantan attrbuton to te autor(s) and te ttle of te work journal ctaton and DOI. Publsed under lcence by IOP Publsng Ltd

3 t Internatonal Conference on "Mes metods for boundary-value problems and applcatons" IOP Publsng IOP Conf. Seres: Materals Scence and Engneerng 58 (6) 4 do:.88/ x/58//4 ere ' 3 b b ( ) ( t) t dt b b (4) ( ) b b (5) 4 5 ( '( ) )' (6) b b b b b b We assume functons k ( x )( ) k( x ) are contnuous wt respect to x are measurable wt respect to x and satsfy te followng condtons for arbtrary x x R R p j j k ( x ) M M M M p (7) p k x M M3 M M3 ( ) (8) ( k ( x ) k ( x ))( ) (9) k ( x ) M M M M p () p k ( x ) M M M M () p 3 3 ( k ( x ) k ( x ))( ). () rougout te followng to defne te generalzed soluton for te problem () (3) we use normed spaces V V ( ) W( ) obtaned by closng C( ) and C ( ; C ( )) n te followng norms u u u V W ( ) W ( ) p p u u u V Lp W p Lp W p ( ) ( ; ( )) ( ; ( )) u u u u W ( ) V ( ) L( ; L ( )) L ( ; L ( )) and we denote te functonal J suc tat ts value on element v can be defned as ere z V ( ) d J( z( t)) v ( z( t)) v( x) dx ( z( t)) v( s) ds Fv s a value of functonal F In papers [] [3] we proved tat space V dt V ( ) (3) ( V ) on te element. concdes wt te set of functons from W p ( ) and ter ncrement on fromw p ( ). Also n papers [] [3] te smlar result was obtaned for V( ).

4 t Internatonal Conference on "Mes metods for boundary-value problems and applcatons" IOP Publsng IOP Conf. Seres: Materals Scence and Engneerng 58 (6) 4 do:.88/ x/58//4 Defnton. Functon u W ( ) suc tat u( x) u ( x) a.e. n and J( u) dt( V ( )) (4) s called generalzed soluton of te problem () (3) f te ntegral dentty v J ( u) v dt k ( x u u) dxdt x u v k x u dsdt f v dt f v dt ( ) s s (5) olds for arbtrary v W ( ) ; ere f v ( f v ) s te value of functonal f from te space L ( ; W ( )) ( and from L p p ( ; W ( )) respectvely) on element v from W( ). p p s paper s a contnuaton of te nvestgatons wc ave been begun n papers [4] [5] were was gven te generalzed statement of te consderng problem te teorems of exstence and unqueness of generalzed soluton ave been proved.. Descrpton of te approxmate metod. e convergence nvestgaton. For problem () (3) we construct te approxmate soluton metod wt respect to varable t and FEM wt respect to space varables. o ts end on [] we defne a unform grd { N } were s te grd ncrement. Under te assumpton tat s a convex doman we defne te polygon nscrbed n wt boundary. s polyedron satsfes te followng condtons: ) for any pont x tere exsts a pont lyng from t at a dstance not more tan ( s te grd ncrement wt respect to space varables); ) ntersecton ponts of and are te vertexes of polygon. e trangulaton of doman s carred out te followng way. For eac of te subdomans and on wc te secton dvdes te doman te trangulaton s performed autonomously suc tat te sets of mes nodes constructed on and and lyng on concde. Suc fragmentaton n negborood of secton provdes us trangles wt one curvlnear sde. Let te V l be a set of te functons from V suc tat ter restrcton on eac fnte element of te trangulaton s an mage of a polynomal of degree l on te basc element (see [6] p.3 [7] p. 47). Defnton. A functon y() t l s called semdscrete problem soluton f for arbtrary z l and t{ } te followng condtons are satsfed V V z y t z x k x y y x dx ( t( ( )) ( ) ( ) ( )) x ( ( ( )) ( ) ( y ) z ( )) t y t z s k x y s ds (6) s s 3

5 t Internatonal Conference on "Mes metods for boundary-value problems and applcatons" IOP Publsng IOP Conf. Seres: Materals Scence and Engneerng 58 (6) 4 do:.88/ x/58//4 Here t f ( x t) f( x ) d. t f ( x t) z( x) dx f ( s t) z( s) ds e followng statements were proved. Lemma. Let te condtons (4) () be satsfed. In addton ncrements were y( x) u ( x) a.e.n and. (7) f L ( ; W ( )) f L ( ; W ( )) u V L ( ) p p p p and are pcked te way tat te followng nequalty olds { } c (8) max p ( p ) p max { p p } p ( p ) max { p } p p p max p p p. { } en te followng pror estmates old for te soluton of problem (6) (7) t y( t) y( t) c (9) L ( ) ( ) L [ ] t p p t W p( ) W p( ) y( t) y( t) c () t t t y ( t) c () t t L ( ) y ( t) c t () t L ( ) k ( ( y( t k )) ( y( t)) )( y( t k ) y( t) ) dx k c (3) k N. Lemma. Let y be a sequence of functons for wc a pror estmates (9) (3) are vald. en tere exsts te functon u W ( ) followng lmt relatons as : and ncrement sequences and tat satsfy te y( t) u n V ( ) (4) 4

6 t Internatonal Conference on "Mes metods for boundary-value problems and applcatons" IOP Publsng IOP Conf. Seres: Materals Scence and Engneerng 58 (6) 4 do:.88/ x/58//4 y( t) u -weakly n L ( ; L ( )) (5) y( t) u -weakly n L ( ; L ( )) (6) were y( t) u a.e. n Q and (7) y( x t) ( x t) y( x t) ( xt ) ( \ ) z z are te pecewse constant extensons of te mes functon z. ey are equal to value zt ( ) on te sets [ t t ][ t t ] respectvely t. Let s consder some proof steps of Lemma. A pror estmates (9) () provde us boundedness of sequences { yt ( )} n V( ) L ( ; L ( )) and n L ( ; L ( )). at s wy tere exst functon u from W( ) and subsequence of sequences { }{ } suc tat as and relatons (4) (6) old. en we take subsequences { }{ } suc tat a pror estmates (9) (3) te compactness teorem (see [8] lemmas.8.9) and (4) (6) provde us te lmt relaton (7). And relatons (8) (9) follow from te estmate () and nequaltes (7) () k ( x y y) K n L ( Q ) (8) p y k( x y ) K n Lp ( ). s e next we proved tat te lmt functon u satsfes () (3). For tat we perform te passage of te lmt n te followng equaton ( y( t)) v dxdt ( u ) v () dx t ( y( t)) v dsdt ( u ) v () ds t v v k x y y dxdt k x y dsdt y ( ) ( ) x s s f v dt f v dt wc s obtaned from (6) as z v ( v s an nterpolant from l constructed for te functon v C( ) C ( ): ( ) ) after te transformaton of te frst and te trd summands wt t elp of te summaton by parts formula and te use of extenson wt respect to varable t and ntegraton over []. As a result we obtan te followng equaton as d v ( u) v dxdt ( u ) v() dx K dxdt x dt V (9) 5

7 t Internatonal Conference on "Mes metods for boundary-value problems and applcatons" IOP Publsng IOP Conf. Seres: Materals Scence and Engneerng 58 (6) 4 do:.88/ x/58//4 d v ( u) v dsdt ( u ) v() ds K dsdt s (3) dt f v dt f v dt. In consderaton of C ( ) we rewrte (3) n te form ( u) vdx ( u) vds dt f v { } { t v v K dx f v K ds dt G( t) dt. } (3) x s From te equaton (3) follows tat functon Gt () s a generalzed dervatve from L( ) of te functon from t follows tat (see (3)) ( u) vdx ( u) vds v v J( u( t)) v f v K dx f v K ds s x almost everywere n []. Next we use te monotone metod to determne tat By usng t we prove tat as suc tat (3) v v { } x s K dx K ds dt v u v { } x s s k ( x u u) dx k ( x u ) ds dt. t t ( ˆ y( t) y( t)) k ( x y y) dxdt x ( ˆ y( t) y( t)) k ( x y y) dsdt s were s from lemma y ˆ( t) y( t ). e man result of ts paper s eorem. Let be a convex doman n te space condtons (9) (3) and let / (33) R functons f L ( ; W ( )) f L ( ; W ( )) u V L ( ) p p p p k ( ) satsfy k 6

8 t Internatonal Conference on "Mes metods for boundary-value problems and applcatons" IOP Publsng IOP Conf. Seres: Materals Scence and Engneerng 58 (6) 4 do:.88/ x/58//4 and condton (9) s satsfed. en te pecewse constant extensons subsequence of sceme soluton (6) (7) wc s satsfyng relatons (4) (9) converges to generalzed soluton of te problem () (3). In condtons of soluton unqueness (see [5]) te wole sequence as ts property. Acknowledgments e researc was supported by te Russan Foundaton for researc (projects ). References [] Antontsev S N and Meyrmanov A M 979 Matematceske model covmestnogo dvgenya povernostny I podzemny vod (Novosbrsk: Novosbrsk State Unversty publsng) [] merbaev M R 4 Spaces wt a grap norm and strengtened Sobolev spaces I Russan Matmatcs [3] merbaev M R 4 Spaces wt a grap norm and strengtened Sobolev spaces II Russan Matmatcs [4] Glazyrna L L and Pavlova M F 994 On solvablty of certan problem on surface and ground water jont moton Russan Matmatcs [5] Glazyrna L L and Pavlova M F A unqueness teorem for te soluton of a problem n te teory of te jon moton of cannel and underground Russan Matmatcs [6] Dautov R Z and Karcevsk M M Vvedene d teoryu metoda konecny elementov (Kazan: Kazan State Unversty publsng) [7] Korneev V G 977 Semy metoda konecny elementov vysok poryadkov tocnost (Lenngrad: Lengrad State Unversty publsng) [8] Alt H.W. and Luckaus S. 983 Quaslnear ellptc-parabolc dfferental equaton Mat.Z