[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85562-en":3,"doc-seo-85562-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85562,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","A Time-Nonlocal Multiphysics Finite Element Method with the Crank-Nicolson Scheme for the Poroelasticity Model with Secondary Consolidation","A time-nonlocal multiphysics finite element method is developed for a poroelasticity model with secondary consolidation using the Crank–Nicolson scheme. Assuming finite positive material parameters, auxiliary variables for fluid content and generalized pressure reformulate the strongly coupled system into a generalized Stokes equation with time-integral terms plus a diffusion equation. A high-order Taylor–Hood spatial discretization and composite trapezoidal quadrature enable efficient real-time updates while preserving second-order temporal accuracy. Weak-solution existence and uniqueness, stability, and optimal error bounds are proven, and numerical experiments compare long-time convergence with backward Euler.","arXiv :2604 .06815v2 [math .NA] 11 Jul 2026  \nA TIME-NONLOCAL MULTIPHYSICS FINITE ELEMENT METHOD WITH THE CRANK-NICOLSON SCHEME FOR THE POROELASTICITY MODEL WITH  \nSECONDARY CONSOLIDATION∗  \nYANAN HE† AND ZHIHAO GE‡  \nAbstract. This paper studies a time-nonlocal multiphysics finite element method with the Crank-Nicolson scheme for the poroelasticity model with secondary consolidation. For the case where the physical parameters λ, λ∗ and c 0 are all finite positive constants, by introducing two auxiliary variables—the fluid content η and the generalized pressure ξ—the original strongly coupled poroelasticity model is reformulated into a generalized Stokes equation with time integral terms and a diffusion equation. The reformulated model not only reveals the underlying multiphysics processes in the original model, but also exhibits time-nonlocal characteristics. A time-nonlocal multiphysics finite element method is designed for thereformulated model: the spatial discretization employs the high-order Taylor-Hood mixed finite element method, and the temporal discretization adopts the Crank-Nicolson scheme. The time integral terms are approximated using the composite trapezoidal rule, and the integral terms Jnξ and Jnη are introduced for real-time updates, which not only avoids repeated calculations and improves efficiency, but also maintains second-order temporal accuracy. The existence and uniqueness of weak solutions for the reformulated model are proved via energy estimate methods, the stability of the fully discrete timenonlocal multiphysics finite element method is established, and optimal-order error estimates are derived using projection operator techniques. Finally, numerical examples verify the theoretical results and compare the long-time convergence of the Crank-Nicolson scheme and the backward Euler scheme.  \nKey words. Poroelasticity model; Multiphysics finite element method; Crank-Nicolson scheme; Stability analysis; Error estimates  \nAMS subject classifications. 65M12, 65M15, 65N30 .  \n1. Introduction. Poroelasticity theory is an important theory for studying the coupling mechanism between solid skeleton deformation and fluid seepage in porous media. It has wide applications in geotechnical engineering, biomedical engineering, energy development, and environmental science.  \nIn geotechnical engineering, the secondary consolidation effect is one of the main causes of long-term settlement of soft soil foundations. The poroelasticity model with secondary consolidation can accurately predict the creep settlement of high-speed railway soft soil subgrades, long-span bridge soft foundations, and high-rise building foundations [1, 2] . It is also used to evaluate the long-term stability of dams, tunnels, tailings ponds, and other engineering structures [3, 4] .  \nIn biomedical engineering, biological soft tissues (such as articular cartilage, intervertebral discs, etc.) have significant viscoelastic characteristics, and their mechanical responses show time-dependent and creep behavior. The poroelasticity model with secondary consolidation can better describe the deformation law of articular cartilage tissue under long-term loading, providing a theoretical tool for studying the pathogenesis of arthritis and intervertebral disc degeneration [5–8] .  \nIn energy resources, the secondary consolidation effect has important influences in long-term oil and gas reservoir exploitation and carbon dioxide geological sequestration projects. The creep deformation of reservoir rocks during long-term production affects wellbore integrity and permeability evolution [9, 10], and directly affects the evaluation results of sequestration safety [11] .  \nIn environmental science, rocks surrounding nuclear waste geological repositories undergo creep deformation under long-term thermal-hydro-mechanical coupling effects [12, 13] . The secondary consolidation effect also affects the long-term performance and remediation efficiency of soil remediation ","cbCailcCm2ndIHjv","https://ap.wps.com/l/cbCailcCm2ndIHjv","pdf",672488,2,1,34,"English","en",105,"# Introduction\n## Problem formulation and modeling background\n# Reformulated model and numerical method\n## Variable reformulation and auxiliary equations\n## Spatial-temporal discretization and quadrature","[{\"question\":\"What is the main goal of this study on poroelasticity with secondary consolidation?\",\"answer\":\"To construct and analyze a time-nonlocal multiphysics finite element method for the poroelasticity model with secondary consolidation, using the Crank–Nicolson time discretization.\"},{\"question\":\"How does the paper reformulate the original strongly coupled model?\",\"answer\":\"By introducing auxiliary variables for fluid content and generalized pressure, the coupled poroelasticity system is rewritten into a generalized Stokes equation with time-integral terms and a diffusion equation.\"},{\"question\":\"What numerical and theoretical results are provided?\",\"answer\":\"The method uses Taylor–Hood mixed finite elements and Crank–Nicolson time stepping with trapezoidal approximation of integral terms; the paper proves weak-solution existence/uniqueness, stability of the fully discrete scheme, and optimal-order error estimates, then verifies them with numerical experiments comparing long-time convergence to backward Euler.\"}]",1784204600,86,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-time-nonlocal-multiphysics-finite-element-method-with-the-crank-nicolson-scheme-for-the-poroelasticity-model-with-secondary-consolidation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-time-nonlocal-multiphysics-finite-element-method-with-the-crank-nicolson-scheme-for-the-poroelasticity-model-with-secondary-consolidation/85562/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","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 is the main goal of this study on poroelasticity with secondary consolidation?","Question",{"text":75,"@type":76},"To construct and analyze a time-nonlocal multiphysics finite element method for the poroelasticity model with secondary consolidation, using the Crank–Nicolson time discretization.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper reformulate the original strongly coupled model?",{"text":80,"@type":76},"By introducing auxiliary variables for fluid content and generalized pressure, the coupled poroelasticity system is rewritten into a generalized Stokes equation with time-integral terms and a diffusion equation.",{"name":82,"@type":73,"acceptedAnswer":83},"What numerical and theoretical results are provided?",{"text":84,"@type":76},"The method uses Taylor–Hood mixed finite elements and Crank–Nicolson time stepping with trapezoidal approximation of integral terms; the paper proves weak-solution existence/uniqueness, stability of the fully discrete scheme, and optimal-order error estimates, then verifies them with numerical experiments comparing long-time convergence to backward Euler.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]