Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations

Construction of Preconditioners by Mapping Properties for Systems of Partial Differential Equations

Author(s): Kent-Andre Mardal, Ragnar Winther

Pp: 66-83 (18)

DOI: 10.2174/978160805291211101010066

* (Excluding Mailing and Handling)

Abstract

The purpose of this chapter is to discuss a general approach to the construction of preconditioners for the linear systems of algebraic equations arising from discretizations of systems of partial differential equations. The discussion here is closely tied to our earlier paper [1], where we gave a comprehensive review of a mathematical theory for constructing preconditioners based on the mapping properties of the coefficient operator of the underlying differential systems. In the presentation given below we focus more on specific examples, while just an outline of the general theory is given.


Keywords: partial differential equations, discretizations, linear systems, algebraic, preconditioners, mathematical theory, Hilbert space, operator

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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 860 and `b`.`banner_from_date` <= 1739834914 and `b`.`banner_to_date` >= 1739834914 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.03msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 860
      • 3: 1739834914
      • 4: 1739834914
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='b3aee4b309370e35934ff6a765108e20'
      190μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    8Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        9ehv1i2FRYOby8w2RGwcYhO5vH27JZuNcdrAxGiC
        uc_cart_id
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        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/933" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
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        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
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        request_cookies
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        ClearShow all
        Date ↕MethodURLData
        #12025-02-17 23:28:34GET/chapter/9331121810