[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84856-en":3,"doc-seo-84856-105":29,"detail-sidebar-cat-0-en-105":91},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},84856,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Adaptive Space-Time Discretized Linear Iterative Scheme for Doubly-Degenerate Parabolic Problems","Degenerate diffusion problems switch the parabolic equation into an ODE or an elliptic equation, often featuring free boundaries that demand extremely small local mesh sizes and time steps. An adaptive space–time formulation is presented using an efficient splitting of nonlinearities. A proposed iterative linearization converts the nonlinear task into a sequence of heat equations, with unconditional convergence and linear rates under non-degeneracy. Robust, fully computable a posteriori estimates are derived by decomposing the nonlinear residual into linearization and discretization errors, enabling fully adaptive solvers validated numerically in 1D and 2D.","arXiv :2607 .05714v1 [math .NA] 7 Jul 2026  \nAn adaptive, space-time discretized linear iterative scheme for doubly-degenerate parabolic problems  \nA. Javed 1 , K. Mitra2 , and I.S. Pop 1  \n1 Hasselt University, Belgium  \n2 Eindhoven University of Technology, The Netherlands  \nJuly 8, 2026  \nAbstract  \nDegenerate diffusion problems, where the governing parabolic equation can change type to either an ordinary differential equation or an elliptic equation, model many real life applications. Due to the presence of free-boundaries, accurate numerical simulation of such problems require extremely small mesh and time step sizes locally. To remediate this issue, in this work, we consider a space-time formulation of the problem based on an efficient splitting of the nonlinearities. First, an iterative linearization scheme is proposed to resolve thenonlinearities that effectively reduces to solving a sequence of heat equations. Unconditional convergence of the scheme is proven even for double degenerate cases with linear convergence achieved if the problem is non-degenerate. Next, the dual norm of the nonlinear residual is decomposed into a linearization error component and a discretization error component corresponding to the heat equation. This leads to reliable and fully computable a posteriori estimates for the problem that are robust with respect to the nonlinearities/degeneracies. These estimates are used then in a fully adaptive (discretization + linearization) space-timesolver. Numerical experiments for multiple test cases (one and two dimensions in space) demonstrate that this solver efficiently allocates the computational resources in the spacetime domain, resulting in a rapid decay of error in terms of total degrees of freedom spent.  \n1 Introduction  \nDegenerate parabolic equations arise in a wide range of applications, such as flow through porous media [4], mathematical biology [22], phase transition [5], traffic flow [53], and other nonlinear diffusion processes. The solution to such models features free boundaries (where the equation becomes an ordinary differential equation(ODE)) with sharp and moving interfaces, causing the solution to have low space regularity, along with degenerate regions (where the equation becomes elliptic) with low time regularity. These two ”degeneracies” make it challenging to numerically approximate the solutions. They also render uniform discretizations (particularly, in time) inefficient since the sharp interfaces often require very fine meshes.  \nLet Ω be a bounded open domain in Rd having a Lipschitz boundary ∂Ω . For some final time T > 0, let Q := (0, T] 􀀂 Ω and Γ := (0, T] 􀀂 ∂Ω be the space-time domain and interface  \nSpace-time iterative scheme and adaptivity Javed, Mitra, Pop  \nrespectively. Then the degenerate diffusion problem reads: find u, w : Q ! R satisfying  \n ∂ tu = ∆w + f , in Q,  \n>  \n ww  Φ0,(u ), inon QΓ,, (1.1)  \n>: u (0) = u0, in Ω,  \nHere, Φ : [0, ω) ! R denotes a nonlinear, monotonically increasing mapping for either ω = 1 or ∞ . In particular, Φ0 might vanish at 0, making the problem an ODE (see Figure 1) . On the other hand, Φ0 might blow up at ω = 1 (for instance, for biofilm growth [22] and traffic flow models [53]) or at infinity, as in the porous medium equation (PME) case. It might even become multivalued at ω = 1, like in the Richards equation case [44] . These scenarios set off the elliptic degeneracy.  \nFigure 1: Examples of degenerate nonlinearities Φ: (Left) Richards-type [44],(Center) biofilmtype [22, 53], and (Right) porous media equation-type [4] degeneracies.  \nTo deal with the double degeneracy discussed above, we use the framework discussed in [1] and also see in [8] . The system is reformulated in terms of a new variable s, and two increasing functions b, B 2 C1 (R) that satisfy  \nΦ = B 􀀎 b􀀀1, and 0 􀀔 b0 , B0 􀀔 1, and b0 + B0 􀀕 1. (1 .2)  \nWith this choice, if u = b (s) and w = B (s), then one immediately gets w 2 Φ (u ) . In this way, problem (1.1) becom","cbCait3lSlnmY0nL","https://ap.wps.com/l/cbCait3lSlnmY0nL","pdf",21806526,1,43,"English","en",105,"# Introduction\n## Degenerate parabolic equations and applications\n## Problem formulation in space-time\n## Reformulation via a new variable\n## Motivation for space-time discretization and adaptivity","[{\"question\":\"What makes doubly-degenerate parabolic problems difficult to simulate numerically?\",\"answer\":\"The governing equation can change type, becoming an ODE or an elliptic equation, which creates free boundaries with sharp moving interfaces and degenerate regions. These features reduce solution regularity and force very small local mesh and time step sizes for accuracy.\"},{\"question\":\"How does the paper reduce the nonlinear problem to linear heat-equation subproblems?\",\"answer\":\"It introduces a space–time formulation and proposes an iterative linearization scheme that splits the nonlinearities effectively. The nonlinearities are resolved by solving a sequence of heat equations, instead of tackling a fully nonlinear system directly.\"},{\"question\":\"What guarantees and error indicators are developed to support adaptivity?\",\"answer\":\"The scheme’s unconditional convergence is proven, with linear convergence under non-degenerate conditions. A posteriori estimates are then constructed by decomposing the dual norm of the nonlinear residual into a linearization error component and a discretization error component tied to the heat equation, enabling fully computable and robust adaptivity.\"}]",1784198835,108,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"adaptive-space-time-discretized-linear-iterative-scheme-for-doubly-degenerate-parabolic-problems","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/adaptive-space-time-discretized-linear-iterative-scheme-for-doubly-degenerate-parabolic-problems/84856/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What makes doubly-degenerate parabolic problems difficult to simulate numerically?","Question",{"text":75,"@type":76},"The governing equation can change type, becoming an ODE or an elliptic equation, which creates free boundaries with sharp moving interfaces and degenerate regions. These features reduce solution regularity and force very small local mesh and time step sizes for accuracy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper reduce the nonlinear problem to linear heat-equation subproblems?",{"text":80,"@type":76},"It introduces a space–time formulation and proposes an iterative linearization scheme that splits the nonlinearities effectively. The nonlinearities are resolved by solving a sequence of heat equations, instead of tackling a fully nonlinear system directly.",{"name":82,"@type":73,"acceptedAnswer":83},"What guarantees and error indicators are developed to support adaptivity?",{"text":84,"@type":76},"The scheme’s unconditional convergence is proven, with linear convergence under non-degenerate conditions. A posteriori estimates are then constructed by decomposing the dual norm of the nonlinear residual into a linearization error component and a discretization error component tied to the heat equation, enabling fully computable and robust adaptivity.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]