[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-227908-en":3,"doc-seo-227908-105":30,"detail-sidebar-cat-0-en-105":97},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},227908,2336478503145,"Aladdin","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","A Mesh-parameter -continuation Method - Preprint - September","A robust method for solving a nonlinear equation is developed, combining a predictor-corrector continuation (PCC) framework with a mesh-continuation strategy. A new stepsize strategy is proposed to improve stability and efficiency when advancing the continuation parameter t. Numerical experiments are presented for a 1-D semiconductor model problem, and guidance is provided for extending the approach to a 2-D implementation, including the handling of residual problems across meshes.","stichtingmathematischcentrum  \nAFDELING NUMERIEKE WISKUNDE(DEPARTMENT OF NUMERICAL MATHEMATICS)  \nNW 89/80  \nSEPTEMBER  \nS.J.POLAK,A.WACHTERS,Th.BEELEN &P.W.HEMKER  \nA MESH-PARAMETER -CONTINUATION METHOD  \nPreprint  \nPrinted at the Mathematical Centre,413 Kruislaan,Amsterdam.  \nThe Mathematical Centre,founded the 11-th o6 February 1946,is a non-progit institution aiming at the promotion of pwre mathematics and itsapplications.It is sponsored by the Netherlands Govenment through theNethereands Onganization {on the Advancement o6 Pure Res eazch(Z.W.0.).  \n\n| 1980:Mathematics   |  |  subject classification:65N99   |\n| --- | --- | --- |\n\nA Mesh -parameter -continuation methoa*)  \nby  \n**)  \n**)  \n**)  \nS.J.Polak,A.Wachters,Th.Beelen &P.W.Hemker  \n# ABSTRACT\n\nIn this report a robust method for the solution of a non-linear equa-  \ntion is considered.A new stepsize strategy for the predictor-corrector  \ncontinuation(PCC-)method is presented.The idea of mesh-continuation is in-  \ntroduced and applied in combination with the PCC-method.Numerical results  \nare shown for a 1-D semiconductor model problem and a suggestion for a  \n2-D implementation is given.  \nKEY WORDS &PHRASES:nonlinear problem,continuation method,mesh continua-tiation  \n# I,INTRODUCTION\n\nThe most important aspect in the construction of program packages forP.D.E.problems [1-3]is Robustness.Other criteria(in order of impor-tance)are user-friendliness and speed.This paper describes our presentefforts to construct Robust algorithms for the solution of a class of non-linear P.D.E.problems.  \nThe algorithms are only considered with respect to a special class ofproblems(section 2)but extensions to other problems are obvious.The pre-dictor -corrector -continuation(PCC)method [4]is one of the most Ro-bust algorithms known to us for the solution of nonlinear equations.Combinations of the PCC method and spatial discretizations are thereforeobvious choices when constructing Robust algorithms for P.D.E.problems[5].Two different combinations are considered in this paper.First a con-tinuation parameter tis introduced in the continuous problem to solve the  \nproblem on a single given mesh(section 4).Secondly a PCC method is usedfor the solution of the residual problem arising from a two mesh algorithm(section 5).In section 6 we consider the simultaneous use of both algo-rithms.In section 7 a practical implementation of this combination isdescribed.  \n# II.SEMICONDUCTOR PROBLEMS\n\nThe equations involved in the analysis of semiconductor problems arediscussed in detail in [6],[7]and [3].Here we only consider the Poissonproblem for the case of negligible currents.In [3]this problem is treatedfor physically realistic 2-D composite regions.  \nIn this paper we use only a 1-D simplified problem as an example.The equation has the form  \n(1)  \n-△u =H(u),  \nwhere  \nwith  \nx∈[20μ,100μ],μ=10**-6,u(20μ)=0,u(100μ)=700,a =10**13,b=40,c₁=0,c₂=700 andD(x)=-a if x\u003C50μ,D(x)=a if x>50μ.  \nThe problem will also be denoted by L(u)=0.There exists a unique boundedsolution for this problem.This is also the case for the more general prob-lem treated in [3].To understand this,a theorem from [8]is used.Thebasic property there is monotony.Our operator is monotonic because -△ iscoercive and)is negative definite.  \nThe upper and lower bound of the solution of the problem are  \nandexp(brespectively,as can be easi-ly shown by techniques similar to those used for the proof of a maximumprinciple [10].These bounds imply that,within the range of floating pointrepresentable numbers,the solution of(1)is a monotonous function.In the general case,a priori lower and upper bounds for the solution are  \nknown on physical grounds.Therefore,a,c₁and c₂are always such that,within rounding errors,;hence,in(1)the function exp(x)may be replaced by 1+x for x≥0.This does not change the solution of theproblem and it eliminates the problems caused by the large exponentialfactor b.  \n# III.THE PCC METHOD\n\nThe PCC method is described in [3]an","cbCaijJlGBPmFf0O","https://ap.wps.com/l/cbCaijJlGBPmFf0O","pdf",1151502,1,11,"English","en",105,"# ABSTRACT\n# I. INTRODUCTION\n# II. SEMICONDUCTOR PROBLEMS\n# III. THE PCC METHOD\n# IV. PARAMETER-CONTINUATION (PC)","[{\"question\":\"What is the main idea of the proposed method?\",\"answer\":\"The report introduces a mesh-continuation strategy combined with a predictor-corrector continuation (PCC) method, using an improved stepsize strategy for advancing the continuation parameter t.\"},{\"question\":\"How is the continuation parameter t used in the formulation?\",\"answer\":\"Parameter t is introduced to modify the problem so that the solution can be tracked from a simpler case (t=0) to the target nonlinear case (t=1), forming an equation F(u(t), t)=0.\"},{\"question\":\"What problems and dimensions are covered by the numerical results?\",\"answer\":\"Results are shown for a 1-D semiconductor model problem, and the paper provides a suggestion for how to implement the approach in 2-D.\"}]","A Mesh-parameter -continuation Method - Preprint - September | 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is the main idea of the proposed method?","Question",{"text":81,"@type":82},"The report introduces a mesh-continuation strategy combined with a predictor-corrector continuation (PCC) method, using an improved stepsize strategy for advancing the continuation parameter t.","Answer",{"name":84,"@type":79,"acceptedAnswer":85},"How is the continuation parameter t used in the formulation?",{"text":86,"@type":82},"Parameter t is introduced to modify the problem so that the solution can be tracked from a simpler case (t=0) to the target nonlinear case (t=1), forming an equation F(u(t), t)=0.",{"name":88,"@type":79,"acceptedAnswer":89},"What problems and dimensions are covered by the numerical results?",{"text":90,"@type":82},"Results are shown for a 1-D semiconductor model problem, and the paper provides a suggestion for how to implement the approach in 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